Lower bound of computational complexity of knapsack problems

Zhidong Zhang; Zhidong Zhang

doi:10.3934/math.2025538

AIMS Mathematics

2025, Volume 10, Issue 5: 11918-11938. doi: 10.3934/math.2025538

Previous Article Next Article

Research article Special Issues

Lower bound of computational complexity of knapsack problems

Zhidong Zhang ^,

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China

Received: 10 January 2025 Revised: 28 April 2025 Accepted: 09 May 2025 Published: 22 May 2025
MSC : 82B20, 82B44, 68Q17, 68Q15

The quantum statistics mechanism is very powerful for investigating the equilibrium states and the phase transitions in complex spin disorder systems. The spin disorder systems act as an interdisciplinary platform for solving the optimum processes in computer science. In this work, I determined the lower bound of the computational complexity of knapsack problems. I investigated the origin of nontrivial topological structures in these hard problems. It was uncovered that the nontrivial topological structures arise from the contradictory between the three-dimensional character of the lattice and the two-dimensional character of the transfer matrices used in the quantum statistics mechanism. I illustrated a phase diagram for the non-deterministic polynomial (NP) vs polynomial (P) problems, in which a NP-intermediate (NPI) area exists between the NP-complete problems and the P-problems, while the absolute minimum core model is at the border between the NPI and the NP-complete problems. The absolute minimum core model of the knapsack problem cannot collapse directly into the P-problem. Under the guide of the results, one may develop the best algorithms for solving various optimum problems in the shortest time (improved greatly from O(1.3^N) to O((1+ε)^N) with ε→0 and ε≠1/N) being in subexponential and superpolynomial. This work illuminates the road on various fields of science ranging from physics to biology to finances, and to information technologies.

Keywords:

Citation: Zhidong Zhang. Lower bound of computational complexity of knapsack problems[J]. AIMS Mathematics, 2025, 10(5): 11918-11938. doi: 10.3934/math.2025538

Related Papers:

[1]	Aslı Alkan, Halil Anaç . The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237
[2]	Musawa Yahya Almusawa, Hassan Almusawa . Numerical analysis of the fractional nonlinear waves of fifth-order KdV and Kawahara equations under Caputo operator. AIMS Mathematics, 2024, 9(11): 31898-31925. doi: 10.3934/math.20241533
[3]	Muath Awadalla, Abdul Hamid Ganie, Dowlath Fathima, Adnan Khan, Jihan Alahmadi . A mathematical fractional model of waves on Shallow water surfaces: The Korteweg-de Vries equation. AIMS Mathematics, 2024, 9(5): 10561-10579. doi: 10.3934/math.2024516
[4]	Khalid Khan, Amir Ali, Muhammad Irfan, Zareen A. Khan . Solitary wave solutions in time-fractional Korteweg-de Vries equations with power law kernel. AIMS Mathematics, 2023, 8(1): 792-814. doi: 10.3934/math.2023039
[5]	Hayman Thabet, Subhash Kendre, James Peters . Travelling wave solutions for fractional Korteweg-de Vries equations via an approximate-analytical method. AIMS Mathematics, 2019, 4(4): 1203-1222. doi: 10.3934/math.2019.4.1203
[6]	Ihsan Ullah, Aman Ullah, Shabir Ahmad, Hijaz Ahmad, Taher A. Nofal . A survey of KdV-CDG equations via nonsingular fractional operators. AIMS Mathematics, 2023, 8(8): 18964-18981. doi: 10.3934/math.2023966
[7]	Musong Gu, Chen Peng, Zhao Li . Traveling wave solution of (3+1)-dimensional negative-order KdV-Calogero-Bogoyavlenskii-Schiff equation. AIMS Mathematics, 2024, 9(3): 6699-6708. doi: 10.3934/math.2024326
[8]	M. Mossa Al-Sawalha, Rasool Shah, Adnan Khan, Osama Y. Ababneh, Thongchai Botmart . Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives. AIMS Mathematics, 2022, 7(10): 18334-18359. doi: 10.3934/math.20221010
[9]	Asif Khan, Tayyaba Akram, Arshad Khan, Shabir Ahmad, Kamsing Nonlaopon . Investigation of time fractional nonlinear KdV-Burgers equation under fractional operators with nonsingular kernels. AIMS Mathematics, 2023, 8(1): 1251-1268. doi: 10.3934/math.2023063
[10]	Yanhua Gu . High-order numerical method for the fractional Korteweg-de Vries equation using the discontinuous Galerkin method. AIMS Mathematics, 2025, 10(1): 1367-1383. doi: 10.3934/math.2025063

Abstract

1. Introduction

The general expansion of integer-order calculus to arbitrary order is fractional calculus (FC), which was developed at the end of the seventeenth century. One of FC's main advantages is that it can be used to describe memory and the inherited characteristics of many different phenomena. Additionally, classical calculus is a small subset of FC. The joint efforts of pioneers like Podlubny ^[1], Ross and Miller ^[2], Caputo ^[3], Riemann ^[4], Liouville ^[5], and many more created the foundation of fractional order derivative. The calculus of fractional order has been used in optics ^[6], electrodynamics ^[7], signal processing ^[8], chaos theory ^[9], and other fields ^[10,11]. It has also been connected to practical initiatives. The nature and behavior of nonlinear issues that emerge in daily life are largely described by the numerical and analytical solutions for differential equations of arbitrary order that arise in the events above ^[12,13].

The solutions of different kinds of nonlinear partial differential equations (NLPDEs) known as solitons exist as a specific form of solitary waves. These superb stable one-wave forms find extensive application in different areas because of their distinct properties. Dispersive waves dominate this system because they both disperse the waves inelastically and result in energy loss from radiation events. Thus, the solitary waves survive by keeping their shape and speed after completing a nonlinear connection. The soliton theory develops essential knowledge about the physical behavior and the meaning of nonlinear events. The theory of solitons attracts scientific exploration because it enables wide-ranging applications within transmission media along with design work, numerical materials study, mathematical physics, and various fields of nonlinear science. Research investigators today show greater interest in obtaining detailed calculations by using expert computation software to reduce complex mathematical operand complexity. Several advanced computational approaches described in the literature explore the characteristics of various types of solutions produced by nonlinear physical models as reported in ^[14,15,16].

Nonlinear systems research has identified fundamental complex dynamics that multiple scientific fields use. Studies in this domain focus on creating a unified model from the Korteweg-de Vries (KdV) equations and Schrödinger. The equartions contain vital elements of wave propagation and soliton processes that make them effective for studying different physical phenomena. When the nonlinear dispersive KdV combines with its quantum mechanical Schrödinger counterpart, it produces a coupled system that contains multi-dimensional linear and nonlinear dynamics processes ^[22,23,24]. The combined system delivers an expanded foundation for studying wave motion along with tools to analyze several remarkable effects created through the combined action of the equations. Scientists study the complete mathematical properties of coupled Schrödinger-KdV equations as a nonlinear system through research conducted from ^[25,26,27].

Analyses of numerous physical systems demand the study of nonlinear systems as a necessary method to understand their behavior. Such nonlinear systems give rise to the linked Schrödinger-KdV equations, which become an outstanding research problem because of their appealing nature. The system incorporates two essential parts containing the KdV equation used to explain nonlinear wave propagation alongside soliton characteristics and the Schrödinger equation dedicated to describing quantum mechanical properties. This combined equation system produces a new system that represents linear quantum phenomena with nonlinear dispersive dynamics, thus making it possible to explore different physical interactions ^[28].

The variational iteration technique was invented by He's ^{[29,30,31,32,33,34]} and has been successfully used in nonlinear poly crystalline solids, self-governing ordinary differential equations ^[34], and other fields. In one degree of freedom, an approximate solution to weakly nonlinear systems is constructed using an averaging technique, variational iteration method, method of variation of constants, and perturbation approach.

Liao Shijun introduced the homotopy analysis technique (HAM) in 1992 ^[35], utilizing homotopy, an essential term in differential geometry and topology. HAM has been successfully used to obtain approximate analytical solutions, free of linearization and perturbation, for issues that emerge in science and technology. Large amounts of computer memory and more time are required for the conventional approach (i.e., HAM). Therefore, combining this method with already-existing transform techniques is necessary to overcome these restrictions. Singh et al. ^[36] introduced the q-homotopy analysis transform method (q-HATM), a sophisticated combination of the Laplace transform and q-HAM. The series solution, which swiftly converges to the precise answer within a brief allowable zone, is managed and controlled by the future scheme. The ability of the suggested method to combine two powerful algorithms to solve linear and nonlinear fractional differential equations analytically and numerically is an improvement. The future approach is free from assumptions, discretization, and perturbation, and it has many robust qualities, such as nonlocal effect, simple solution process, and promising broad convergence region. The Laplace transform using semi-analytical methods takes less computation. It is time to calculate the solutions for complicated nonlinear models and phenomena that occur in research and industry, which is worth disclosing. The two auxiliary parameters, $\hbar$ and $n$ , that the q-HATM solution uses enable us to modify and regulate the solution's convergence. The suggested method can provide outcomes with greater efficiency while requiring less calculation time and effort than conventional methods. Many writers have recently used q-HATM to analyze various phenomena located in various locations. For example, the model of vibration equation of arbitrary order is studied by Srivastava et al. ^[37], the model of Lienard's equation is studied by Kumar et al. ^[40], many authors find the approximated analytical solution for nonlinear problems ^[41,42], Bulut and his coauthors analyze the arbitrary order model of the HIV infection ^[38], and Prakash and his coauthor find the approximated analytical solution for fractional multi-dimensional telegraph equation ^[39]. Many others find the approximated analytical solution for nonlinear problems ^[41,42].

The implementation of the q-HATM and the variational iteration transform method (VITM) to the fractional-order system of Schrödinger-KdV equations is examined in this paper. This study is innovative because it uses an extraordinary transformation known as the Mohand transformation (MT). The solutions to the provided issues demonstrate the efficacy of the proposed tactics. Tables and statistics are used to examine and illustrate the solutions of the recommended approaches.

2. Concepts of Mohand transform

Definition 2.1. Assume that $\mathbb{U}(\xi)$ is the function for which ^[43] defines the MT as:

$M[\mathbb{U}(\xi)] = R(s) = s^2\int_{0}^{\xi}\mathbb{U}(\xi)e^{-s\xi}d{\xi},\ \ {k}_{1}\leq s \leq{k}_{2}.$

A description of the Mohand inverse transform (MIT) appears in the following explanation:

$M^{-1}[R(s)] = \mathbb{U}(\xi).$

Definition 2.2 （^[44]). The fractional MT derivative is given as:

$M[\mathbb{U}^{\sigma}(\xi)] = s^{\sigma}R(s)-\sum\limits_{k = 0}^{n-1}\frac{\mathbb{U}^{k}(0)}{s^{k-(\sigma+1)}},\ 0 < \sigma\leq n.$

Definition 2.3. The following are a few of the features of the MT:

$(1)$ $M[\mathbb{U}'(\xi)] = sR(s)-s^{2}R(0)$ ,

$(2)$ $M[\mathbb{U}''(\xi)] = s^{2}R(s)-s^{3}R(0)-s^{2}R'(0)$ ,

$(3)$ $M[\mathbb{U}^{n}(\xi)] = s^{n}R(s)-s^{n+1}R(0)-s^{n}R'(0)-\cdots-s^{n}R^{n-1}(0)$ .

Lemma 2.1. Assume that $\mathbb{U}({{\eta}}, {\xi})$ is an exponentially ordered function. With $M[R(s)] = \mathbb{U}({{\eta}}, {{\xi}})$ , the MT is then characterized as:

$\begin{equation} M[D_{{{\xi}}}^{r{\sigma}}\mathbb{U}({{\eta}}, {{\xi}})] = {s}^{r{\sigma}}R(s)-\sum\limits_{j = 0}^{r-1}{s}^{{\sigma}(r-j)-1}D_{{{\xi}}}^{j{\sigma}}\mathbb{U}({{\eta}},0), 0 < {\sigma}\leq 1, \end{equation}$

(2.1)

where ${{\eta}} = ({{\eta}}_{1}, {{\eta}}_{2}, \cdots, {{\eta}}_{\sigma})\in \mathbb{R}^{\sigma}$ , $\sigma\in \mathbb{N}$ , and $D_{{{\xi}}}^{r{\sigma}} = D_{{{\xi}}}^{{\sigma}}.D_{{{\xi}}}^{{\sigma}}.\cdots.D_{{{\xi}}}^{{\sigma}}(r-times)$ .

Proof. The induction method demonstrates the truthfulness of Eq (2.1), while $r = 1$ allows solving the equation to obtain the following result:

$M[D_{{{\xi}}}^{2{\sigma}}\mathbb{U}({{\eta}}, {{\xi}})] = {s}^{2{\sigma}}R(s)-{s}^{2{\sigma}-1}\mathbb{U}({{\eta}},0)-{s}^{{\sigma}-1}D_{{{\xi}}}^{{\sigma}}\mathbb{U}({{\eta}},0).$

By Definition 2.2, Eq (2.1) is proven to be true for $r = 1$ . $r = 2$ is substituted in Eq (2.1) to obtain:

$\begin{equation} M[D_{r}^{2{\sigma}}\mathbb{U}({{\eta}}, {{\xi}})] = {s}^{2{\sigma}}R(s)-{s}^{2{\sigma}-1}\mathbb{U}({{\eta}},0)-{s}^{{\sigma}-1}D_{{{\xi}}}^{{\sigma}}\mathbb{U}({{\eta}},0). \end{equation}$

(2.2)

Take Eq (2.2) on the left-hand side:

$\begin{equation} L.H.S = M[D_{{{\xi}}}^{2{\sigma}}\mathbb{U}({{\eta}}, {{\xi}})]. \end{equation}$

(2.3)

Another approach to define Eq (2.3) is as follows:

$\begin{equation} L.H.S = M[D_{{{\xi}}}^{{\sigma}}D_{{{\xi}}}^{{\sigma}}\mathbb{U}({{\eta}}, {{\xi}})]. \end{equation}$

(2.4)

Assume

$\begin{equation} z({{\eta}}, {{\xi}}) = D_{{{\xi}}}^{{\sigma}}\mathbb{U}({{\eta}}, {{\xi}}). \end{equation}$

(2.5)

In Eq (2.4), put Eq (2.5) and then solve to get the following result:

$\begin{equation} L.H.S = M[D_{{{\xi}}}^{{\sigma}}z({{\eta}}, {{\xi}})]. \end{equation}$

(2.6)

In Eq (2.6), the Caputo derivative results in the following modifications:

$\begin{equation} L.H.S = M[J^{1-{\sigma}}z^{'}({{\eta}}, {{\xi}})]. \end{equation}$

(2.7)

The following is the solution of Riemann-Liouville (RL) integral modification to Eq (2.7):

$\begin{equation} L.H.S = \frac{M[z^{'}({{\eta}}, {{\xi}})]}{{s}^{1-{\sigma}}}. \end{equation}$

(2.8)

These results are achieved by using the MT derivative feature on Eq (2.8)

$\begin{equation} L.H.S = {s}^{{\sigma}}Z({{\eta}},{s})-\frac{z({{\eta}},0)}{{s}^{1-{\sigma}}}. \end{equation}$

(2.9)

The solution was achieved by applying Eq (2.5)

$Z({{\eta}},{s}) = {s}^{{\sigma}}R(s)-\frac{\mathbb{U}({{\eta}},0)}{{s}^{1-{\sigma}}}.$

As $M[z({{\xi}}, {{\eta}})] = Z({{\eta}}, {s})$ , consequently, Eq (2.9) may also be written as follows:

$\begin{equation} L.H.S = {s}^{2{\sigma}}R(s)-\frac{\mathbb{U}({{\eta}},0)}{{s}^{1-2{\sigma}}}-\frac{D_{{{\xi}}}^{{\sigma}}\mathbb{U}({{\eta}},0)}{{s}^{1-{\sigma}}}. \end{equation}$

(2.10)

Suppose that for $r = K$ , Eq (2.1) holds. Apply $r = K$ to Eq (2.1), then

$\begin{equation} M[D_{{{\xi}}}^{K{\sigma}}\mathbb{U}({{\eta}}, {{\xi}})] = {s}^{K{\sigma}}R(s)-\sum\limits_{j = 0}^{K-1}{s}^{{\sigma}(K-j)-1}D_{{{\xi}}}^{j{\sigma}}D_{{{\xi}}}^{j{\sigma}}\mathbb{U}({{\eta}},0),\ 0 < {\sigma}\leq1. \end{equation}$

(2.11)

Another step is to provide that Eq (2.1) holds for $r = K+1$ . Now, solve Eq (2.1), by replacing $r = K+1$ .

$\begin{equation} M[D_{{{\xi}}}^{(K+1){\sigma}}\mathbb{U}({{\eta}}, {{\xi}})] = {s}^{(K+1){\sigma}}R(s)-\sum\limits_{j = 0}^{K}{s}^{{\sigma}((K+1)-j)-1}D_{{{\xi}}}^{j{\sigma}}\mathbb{U}({{\eta}},0). \end{equation}$

(2.12)

Equation (2.12) may be used to deduce this result:

$\begin{equation} L.H.S = M[D_{{{\xi}}}^{K{\sigma}}(D_{{{\xi}}}^{K{\sigma}})]. \end{equation}$

(2.13)

Let

$D_{{{\xi}}}^{K{\sigma}} = g({{\eta}}, {{\xi}}).$

The following solution is achieved by applying Eq (2.13):

$\begin{equation} L.H.S = M[D_{{{\xi}}}^{{\sigma}}g({{\eta}}, {{\xi}})]. \end{equation}$

(2.14)

The Caputo's derivative and the RL integral are used to solve Eq (2.14)

$\begin{equation} L.H.S = {s}^{{\sigma}}M[D_{{{\xi}}}^{K{\sigma}}\mathbb{U}({{\eta}}, {{\xi}})]-\frac{g({{\eta}},0)}{{s}^{1-{\sigma}}}. \end{equation}$

(2.15)

Equation (2.11) may be used as a basis to develop Eq (2.15)

$\begin{equation} L.H.S = {s}^{r{\sigma}}R(s)-\sum\limits_{j = 0}^{r-1}{s}^{{\sigma}(r-j)-1}D_{{{\xi}}}^{j{\sigma}}\mathbb{U}({{\eta}},0). \end{equation}$

(2.16)

Here is another way to express Eq (2.16) as:

$L.H.S = M[D_{{{\xi}}}^{r{\sigma}}\mathbb{U}({{\eta}},0)].$

Equation (2.1) holds for $r = K+1$ , as mathematical induction demonstrates. It follows that for any positive integer, Eq (2.1) holds. □

Lemma 2.2. Suppose that the order of the function $\mathbb{U}({{\eta}}, {{\xi}})$ is exponential. $M[\mathbb{U}({{\eta}}, {{\xi}})] = R(s)$ shows the MT of $\mathbb{U}({{\eta}}, {{\xi}})$ . The MT contains the following statement about multiple fractional power series (MFPS):

$\begin{equation} R(s) = \sum\limits_{r = 0}^{\infty}\frac{\hbar_{r}({{\eta}})}{s^{r{\sigma}+1}}, s > 0, \end{equation}$

(2.17)

where ${{\eta}} = (\eta_{1}, {{\eta}}_{2}, \cdots, {{\eta}}_{\sigma})\in\mathbb{R}^{\sigma}, \ \sigma\in \mathbb{N}$ .

Proof. Consider the Taylor series:

$\begin{equation} \mathbb{U}({{\eta}}, {{\xi}}) = \hbar_{0}({{\eta}})+\hbar_{1}({{\eta}})\frac{{{\xi}}^{{\sigma}}}{\Gamma[{\sigma}+1]}++\hbar_{2}({{\eta}})\frac{{{\xi}}^{2{\sigma}}}{\Gamma[2{\sigma}+1]}+\cdots. \end{equation}$

(2.18)

We reach the following result by applying MT to Eq (2.18):

$M\left[\mathbb{U}({{\eta}}, {{\xi}})\right] = M\left[\hbar_{0}({{\eta}})\right]+M\left[\hbar_{1}({{\eta}})\frac{{{\xi}}^{{\sigma}}}{\Gamma[{\sigma}+1]}\right]+M\left[\hbar_{1}({{\eta}}) \frac{{{\xi}}^{2{\sigma}}}{\Gamma[2{\sigma}+1]}\right]+\cdots .$

To achieve the subsequent result, make use of MT's features

$M\left[\mathbb{U}({{\eta}}, {{\xi}})\right] = \hbar_{0}({{\eta}})\frac{1}{s}+\hbar_{1}({{\eta}})\frac{\Gamma[{\sigma}+1]}{\Gamma[{\sigma}+1]}\frac{1}{s^{{\sigma}+1}}+\hbar_{2}({{\eta}}) \frac{\Gamma[2{\sigma}+1]}{\Gamma[2{\sigma}+1]}\frac{1}{s^{2{\sigma}+1}}\cdots .$

This leads to an updated Taylor series. □

Lemma 2.3. In MFPS, the modified Taylor series takes this form when MT is symbolized as $M[\mathbb{U}(\eta, \xi)] = R(s)$

$\begin{equation} \hbar_{0}({{\eta}}) = \lim\limits_{s\rightarrow \infty}sR(s) = \mathbb{U}({{\eta}},0). \end{equation}$

(2.19)

Proof. Consider the Taylor series:

$\begin{equation} \hbar_{0}({{\eta}}) = sR(s)-\frac{\hbar_{1}({{\eta}})}{s^{{\sigma}}}-\frac{\hbar_{2}({{\eta}})}{s^{2{\sigma}}}-\cdots . \end{equation}$

(2.20)

The limit determined in Eq (2.19) is computed and simplified to provide Eq (2.20). □

Theorem 2.4. Let $M[\mathbb{U}({{\eta}}, {{\xi}})]$ denote a function. Then, $R(s)$ is expressed in the MFPS notation as follows:

$R(s) = \sum\limits_{r = 0}^{\infty}\frac{\hbar_{r}({{\eta}})}{{s}^{r{\sigma}+1}}, \ {s} > 0,$

where ${{\eta}} = ({{\eta}}_{1}, {{\eta}}_{2}, \cdots, {{\eta}}_{\sigma})\in\mathbb{R}^\sigma$ and $\sigma\in\mathbb{N}$ . Then, we have

$\hbar_{r}({{\eta}}) = D_{r}^{r{\sigma}}\mathbb{U}({{\eta}},0),$

where $D_{{{\xi}}}^{r{\sigma}} = D_{{{\xi}}}^{{\sigma}}.D_{{{\xi}}}^{{\sigma}}.\cdots.D_{{{\xi}}}^{{\sigma}}(r-times)$ .

Proof. Consider the Taylor series:

$\begin{equation} \hbar_{1}({{\eta}}) = {s}^{{\sigma}+1}R(s)-{s}^{{\sigma}}\hbar_{0}({{\eta}})-\frac{\hbar_{2}({{\eta}})}{{s}^{{\sigma}}}-\frac{\hbar_{3}({{\eta}})}{{s}^{2{\sigma}}}- \cdots. \end{equation}$

(2.21)

By applying limit to Eq (2.21), we obtain the subsequent result:

$\hbar_{1}({{\eta}}) = \lim\limits_{{s}\rightarrow \infty}({s}^{{\sigma}+1}R(s)-{s}^{{\sigma}}\hbar_{0}({{\eta}}))-\lim\limits_{{s}\rightarrow \infty}\frac{\hbar_{2}({{\eta}})}{{s}^{{\sigma}}} -\lim\limits_{{s}\rightarrow \infty}\frac{\hbar_{3}({{\eta}})}{{s}^{2{\sigma}}}- \cdots.$

Simplifying further, we get

$\begin{equation} \hbar_{1}({{\eta}}) = \lim\limits_{{s}\rightarrow \infty}({s}^{{\sigma}+1}R(s)-{s}^{{\sigma}}\hbar_{0}({{\eta}})). \end{equation}$

(2.22)

Using the basic ideas outlined in Lemma 2.1, we may construct the following variant of Eq (2.22):

$\begin{equation} \hbar_{1}({{\eta}}) = \lim\limits_{{s}\rightarrow \infty}({s}M[D_{{{\xi}}}^{{\sigma}}\mathbb{U}({{\eta}}, {{\xi}})]({s})). \end{equation}$

(2.23)

Equation (2.23) is derived using Lemma 2.2 as a foundation:

$\hbar_{1}({{\eta}}) = D_{{{\xi}}}^{{\sigma}}\mathbb{U}({{\eta}},0).$

The following result requires taking $\lim_{{s}\rightarrow \infty}$ once again and applying the Taylor series

$\hbar_{2}({{\eta}}) = {s}^{2{\sigma}+1}R(s)-{s}^{2{\sigma}}\hbar_{0}({{\eta}})-{s}^{{\sigma}}\hbar_{1}({{\eta}})-\frac{\hbar_{3}({{\eta}})}{{s}^{{\sigma}}}- \cdots .$

Using Lemma 2.2, the following results are obtained:

$\begin{equation} \hbar_2({{\eta}}) = \lim\limits_{{s}\rightarrow \infty}{s}({s}^{2{\sigma}}R(s)-{s}^{2{\sigma}-1}\hbar_{0}({{\eta}})-{s}^{{\sigma}-1}\hbar_{1}({{\eta}})). \end{equation}$

(2.24)

Lemmas 2.1 and 2.3 serve as the basis for modifying the Eq (2.24) in the following way:

$\hbar_{2}({{\eta}}) = D_{{{\xi}}}^{2{\sigma}}\mathbb{U}({{\eta}},0).$

By following the same process, we get:

$\hbar_{3}({{\eta}}) = \lim\limits_{{s}\rightarrow \infty}{s}(M[D_{{{\xi}}}^{2{\sigma}}\mathbb{U}({{\eta}},{\sigma})]({s})).$

This final solution is found using Lemma 2.3:

$\hbar_{3}({{\eta}}) = D_{{{\xi}}}^{3{\sigma}}\mathbb{U}({{\eta}},0).$

Generally,

$\hbar_{r}({{\eta}}) = D_{{{\xi}}}^{r{\sigma}}\mathbb{U}({{\eta}},0).$

□

The following theorem describes and explains the new form of Taylor's series convergence.

3. Methodology

3.1. q-homotopy Mohand transform method (q-HMTM)

Consider the following nonhomogeneous, nonlinear fractional order partial differential equation (PDE):

$\begin{equation} \begin{split} &D_{\xi}^{\sigma}{\mathbb{U}}(\eta,\xi)+\mathcal{R}{\mathbb{U}}(\eta,\xi)+\mathcal{N}{\mathbb{U}}(\eta,\xi) = {\mathbb{H}}(\eta,\xi),\ \ n-1 < \sigma\leq n. \end{split} \end{equation}$

(3.1)

$D_{\mathcal{R}}^{\sigma}{\mathbb{U}}(\eta, \xi)$ represents the Caputo derivative, whereas $\mathcal{N}$ and $\mathcal{R}$ represent the nonlinear and linear operators, respectively, and ${\mathbb{H}}(\eta, \xi)$ is the source.

Apply Mohand transform to Eq (3.1)

$\begin{equation} \begin{split} \mathcal{M}[{\mathbb{U}}(\eta,\xi)]-\frac{1}{s^{\sigma}}\sum\limits_{k = 0}^{n-1}s^{\sigma-k-1}{\mathbb{U}}^{k}(\eta,0)+\frac{1}{s^{\sigma}}[\mathcal{M}[\mathcal{R} {\mathbb{U}}(\eta,\xi)]+\mathcal{M}[\mathcal{N}{\mathbb{U}}(\eta,\xi)]-\mathcal{M}[{\mathbb{H}}(\eta,\xi)]] = 0. \end{split} \end{equation}$

(3.2)

The nonlinear function is given as:

$\begin{equation} \begin{split} {N}[{\varphi}(\eta,\xi;q)]& = \mathcal{M}[{\varphi}(\eta,\xi;q)]-\frac{1}{s^{\sigma}}\sum\limits_{k = 0}^{n-1}s^{\sigma-k-1}{\varphi}^{k}(\eta,\xi;q)(0^{+}) +\frac{1}{s^{\sigma}}[\mathcal{M}[\mathcal{R}{\varphi}(\eta,\xi;q)]\\&+\mathcal{M}[\mathcal{N}{\varphi}(\eta,\xi;q)]-\mathcal{M}[{\mathbb{H}}(\eta,\xi)]]. \end{split} \end{equation}$

(3.3)

Here, ${\varphi}(\eta, \xi; q))$ w.r.t $\eta$ , $\xi$ , and $q\in\Big[0, \frac{1}{n}\Big]$ are the real-value functions. A homotopy can be given as:

$\begin{equation} \begin{split} (1-nq)\mathcal{M}[{\varphi}(\eta,\xi;q)-{\mathbb{U}_{0}}(\eta,\xi)] = \hbar q\mathfrak{h}(\eta,\xi)\mathcal{N}[{\varphi}(\eta,\xi;q)]. \end{split} \end{equation}$

(3.4)

In the above expression, the initial condition is denoted by ${\mathbb{U}_{0}}$ , and the auxiliary parameter is denoted by $\hbar = 0$ .

The following result holds for both $0$ and $\frac{1}{n}$ :

$\begin{equation} \begin{split} {\varphi}(\eta,\xi;0) = {\mathbb{U}_{0}}(\eta,\xi),\ \ {\varphi}(\eta,\xi;\frac{1}{n}) = {\mathbb{U}}(\eta,\xi). \end{split} \end{equation}$

(3.5)

The strengthening of the $q$ parameter causes the solution ${\varphi}(\eta, \xi; q)$ to diverge from the initial predictive value ${\mathbb{U}_{0}}(\eta, \xi)$ to ${\mathbb{U}}(\eta, \xi)$ . The following expression results from applying Taylor's theorem to function ${\varphi}(\eta, \xi; q)$ when it depends on variable $q$ :

$\begin{equation} \begin{split} {\varphi}(\eta,\xi;q) = {\mathbb{U}_{0}}(\eta,\xi)+\sum\limits_{{m} = 1}^{\infty}{\mathbb{U}_{{m}}}(\eta,\xi)q^{{m}}, \end{split} \end{equation}$

(3.6)

where

$\begin{equation} \begin{split} {\mathbb{U}_{{m}}} = \frac{1}{{m}!}\frac{\partial^{m}{\varphi}(\eta,\xi;q)}{\partial q^{{m}}}|_{q = 0} \end{split}. \end{equation}$

(3.7)

The convergences series of (3.4) at $q = \frac{1}{n}$ for the appropriate values of ${\mathbb{U}_{0}}(\eta, \xi)$ , $n$ , and $\hbar$ . Consequently,

$\begin{equation} \begin{split} {\varphi}(\eta,\xi;q) = {\mathbb{U}_{0}}(\eta,\xi)+\sum\limits_{{m} = 1}^{\infty}{\mathbb{U}_{{m}}}(\eta,\xi)\Big(\frac{1}{n}\Big)^{{m}}. \end{split} \end{equation}$

(3.8)

The derivatives of Eq (3.4) and the parameter $q$ are} obtained by $q = 0$ , dividing by ${m}!$ , and calculating the derivative

$\begin{equation} \begin{split} \mathcal{M}[{\mathbb{U}_{{m}}}(\eta,\xi)-k_{{m}}{\mathbb{U}_{{m}-1}}(\eta,\xi)] = \hbar\mathfrak{h}(\eta,\xi) \mathcal{R}_{{m}}(\overrightarrow{{\mathbb{U}}}_{{m}-1}). \end{split} \end{equation}$

(3.9)

The auxiliary parameters $\hbar\neq0$ and the vectors are expressed as:

$\begin{equation} \begin{split} \overrightarrow{{\mathbb{U}}}_{{m}} = [{\mathbb{U}_{0}}(\eta,\xi),{\mathbb{U}_{1}}(\eta,\xi),\cdots,{\mathbb{U}_{{m}}}(\eta,\xi)]. \end{split} \end{equation}$

(3.10)

The outcome of applying the MIT to Eq (3.9) is as follows:

$\begin{equation} \begin{split} {\mathbb{U}_{{m}}}(\eta,\xi) = k_{{m}}{\mathbb{U}_{{m}-1}}(\eta,\xi)+\hbar\mathcal{M}^{-1}[\mathfrak{h}(\eta,\xi) \mathcal{R}_{{m}}(\overrightarrow{{\mathbb{U}}}_{{m}-1})], \end{split} \end{equation}$

(3.11)

$\begin{equation*} \begin{split} \mathcal{R}_{{m}}(\overrightarrow{{\mathbb{K}}}_{{m}-1}) = \frac{1}{({m}-1)!}\frac{\partial^{{m}-1}\mathcal{N}[{\varphi}(\eta,\xi;q)]}{\partial q^{{m}-1}}\mid_{q = 0}, \end{split} \end{equation*}$

$\begin{equation} \begin{split} k_{{m}} = \begin{cases}0\ \ \ if \ \ {m}\leq1,\\1\ \ \ if \ \ {m} > 1.\end{cases} \end{split} \end{equation}$

(3.12)

The elements of the q-HATM result are ultimately determined by solving Eq (3.11).

3.2. VITM

Consider the following nonhomogeneous, nonlinear fractional order PDE:

$\begin{equation} \begin{split} &D_{\xi}^{\sigma}{\mathbb{U}}(\eta,\xi) = \mathcal{R}{\mathbb{U}}(\eta,\xi)+\mathcal{N}{\mathbb{U}}(\eta,\xi)+{\mathbb{H}}(\eta,\xi),\ \ n-1 < \sigma\leq n. \end{split} \end{equation}$

(3.13)

Initial condition

$\begin{equation} \begin{split} {\mathbb{U}}(\eta,0) = {\mathbb{U}_{0}}(\eta). \end{split} \end{equation}$

(3.14)

Using the MT in Eq (3.13) is given as

$\begin{equation} \begin{split} &\mathcal{M}[D_{\xi}^{\sigma}{\mathbb{U}}(\eta,\xi)] = \mathcal{M}[\mathcal{R}{\mathbb{U}}(\eta,\xi)+\mathcal{N}{\mathbb{U}}(\eta,\xi) +{\mathbb{H}}(\eta,\xi)]. \end{split} \end{equation}$

(3.15)

The iteration property of transformation is defined as:

$\begin{equation} \begin{split} &\mathcal{M}[{\mathbb{U}}(\eta,\xi)]-\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{U}}(\eta,\xi)} {\partial\xi^{{k}}}|_{\xi = 0} = \mathcal{M}[\mathcal{R}{\mathbb{U}}(\eta,\xi)+\mathcal{N}{\mathbb{U}}(\eta,\xi)+{\mathbb{H}}(\eta,\xi)], \end{split} \end{equation}$

(3.16)

with the Lagrange multiplier of $(-\lambda({s}))$ ,

$\begin{equation} \begin{split} &\mathcal{M}[{\mathbb{U}_{n+1}}(\eta,\xi)] = \mathcal{M}[{\mathbb{U}_{n}}(\eta,\xi)] -\lambda({s})\Big[\mathcal{M}[{\mathbb{U}_{n}}(\eta,\xi)]-\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{U}} (\eta,0)}{\partial\xi^{{k}}}\Big], \end{split} \end{equation}$

(3.17)

where $\lambda({s}) = -\frac{1}{{s}^{\sigma}}$ , and substituting Eq (3.17) into Eq (3.16), we obtain

$\begin{equation} \begin{split} \mathcal{M}[{\mathbb{U}}_{n+1}(\eta,\xi)]& = \mathcal{M}[{\mathbb{U}_{n}}(\eta,\xi)] -\lambda({s})\Big[\mathcal{M}[{\mathbb{U}_{n}}(\eta,\xi)]-\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{U}} (\eta,0)}{\partial\xi^{{k}}}\\&+\mathcal{M}[\mathcal{R}{\mathbb{U}}(\eta,\xi)+\mathcal{N}{\mathbb{U}}(\eta,\xi) +{\mathbb{H}}(\eta,\xi)]\Big]. \end{split} \end{equation}$

(3.18)

The MIT in Eq (3.18) enables us to derive:

$\begin{equation} \begin{split} {\mathbb{U}}_{n+1}(\eta,\xi)& = {\mathbb{U}_{n}}(\eta,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{U}} (\eta,0)}{\partial\xi^{{k}}}\\ &+\mathcal{M}[\mathcal{R}{\mathbb{U}}(\eta,\xi)+\mathcal{N}{\mathbb{U}}(\eta,\xi) +{\mathbb{H}}(\eta,\xi)]\Big]. \end{split} \end{equation}$

(3.19)

The starting constraints become:

$\begin{equation} \begin{split} &{\mathbb{U}}_{0}(\eta,\xi) = \mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{U}} (\eta,0)}{\partial\xi^{{k}}}\Big]. \end{split} \end{equation}$

(3.20)

The recursive system is followed as:

(3.21)

4. Problem 1

4.1. The implementation of q-HATM

Let's assume the system of Schrödinger-KdV equations of time-fractional order:

$\begin{equation} \begin{split} &D_{{{\xi}}}^{\sigma}\mathbb{U}(\eta ,\xi )-\frac{\partial ^2\mathbb{V}(\eta ,\xi )}{\partial \eta^2}-\mathbb{V}(\eta ,\xi )\mathbb{W}(\eta ,\xi ) = 0,\\ &D_{{{\xi}}}^{\sigma}\mathbb{V}(\eta ,\xi )+\frac{\partial ^2\mathbb{U}(\eta ,\xi )}{\partial \eta^2}+\mathbb{U}(\eta ,\xi )\mathbb{W}(\eta ,\xi ) = 0,\\ &D_{{{\xi}}}^{\sigma}\mathbb{W}(\eta ,\xi )+6\mathbb{W}(\eta ,\xi )\frac{\partial \mathbb{W}(\eta ,\xi )}{\partial \eta }+\frac{\partial ^3\mathbb{W}(\eta ,\xi )}{\partial \eta^3}-2\mathbb{U}(\eta ,\xi )\frac{\partial \mathbb{U}(\eta ,\xi )}{\partial \eta}-2\mathbb{V}(\eta ,\xi )\frac{\partial \mathbb{V}(\eta ,\xi )}{\partial \eta } = 0, \end{split} \end{equation}$

(4.1)

where $0 < \sigma \leq 1$ .

Initial conditions are given as:

$\begin{equation} \begin{split} &{\mathbb{U}}({{\eta}},0) = \cos (\eta ) \tanh (\eta ),\\ &{\mathbb{V}}({{\eta}},0) = \sin (\eta ) \tanh (\eta ),\\ &{\mathbb{W}}({{\eta}},0) = \frac{7}{8}-2 \tanh ^2(\eta ), \end{split} \end{equation}$

(4.2)

$\begin{equation} \begin{split} &\mathcal{M}[{\mathbb{U}}(\eta ,\xi )]+{s}\left(\cos (\eta ) \tanh (\eta )\right)+\frac{1}{s^{\sigma}}\mathcal{M}\Big[-\frac{\partial ^2\mathbb{V}(\eta ,\xi )}{\partial \eta^2}-\mathbb{V}(\eta ,\xi )\mathbb{W}(\eta ,\xi )\Big] = 0,\\ &\mathcal{M}[{\mathbb{V}}(\eta ,\xi )]+{s}\left(\sin (\eta ) \tanh (\eta )\right)+\frac{1}{s^{\sigma}}\mathcal{M}\Big[\frac{\partial ^2\mathbb{U}(\eta ,\xi )}{\partial \eta^2}+\mathbb{U}(\eta ,\xi )\mathbb{W}(\eta ,\xi )\Big] = 0,\\ &\mathcal{M}[{\mathbb{W}}(\eta ,\xi )]+{s}\left(\frac{7}{8}-2 \tanh ^2(\eta )\right)+\frac{1}{s^{\sigma}}\mathcal{M}\Big[6\mathbb{W}(\eta ,\xi )\frac{\partial \mathbb{W}(\eta ,\xi )}{\partial \eta }+\frac{\partial ^3\mathbb{W}(\eta ,\xi )}{\partial \eta^3}-2\mathbb{U}(\eta ,\xi )\frac{\partial \mathbb{U}(\eta ,\xi )}{\partial \eta}\\&-2\mathbb{V}(\eta ,\xi )\frac{\partial \mathbb{V}(\eta ,\xi )}{\partial \eta }\Big] = 0. \end{split} \end{equation}$

(4.3)

The definition} of nonlinear operators is given as follows:

$\begin{equation} \begin{split} \mathcal{N}^1[\varphi_1(\eta ,\xi;q),\varphi_2(\eta ,\xi;q),\varphi_3(\eta ,\xi;q)]& = \mathcal{M}[\varphi_1(\eta ,\xi;q)]+{s}\left(\cos (\eta ) \tanh (\eta )\right)\\&+\frac{1}{s^{\sigma}}\mathcal{M}\Big[-\frac{\partial ^2\varphi_2(\eta ,\xi;q)}{\partial \eta^2}-\varphi_2(\eta ,\xi;q)\varphi_3(\eta ,\xi;q)\Big],\\ \mathcal{N}^2[\varphi_1(\eta ,\xi;q),\varphi_2(\eta ,\xi;q),\varphi_3(\eta ,\xi;q)]& = \mathcal{M}[\varphi_2(\eta ,\xi;q)]+{s}\left(\sin (\eta ) \tanh (\eta )\right)\\&+\frac{1}{s^{\sigma}}\mathcal{M}\Big[\frac{\partial ^2\varphi_1(\eta ,\xi;q)}{\partial \eta^2}+\varphi_1(\eta ,\xi;q)\varphi_3(\eta ,\xi;q)\Big],\\ \mathcal{N}^3[\varphi_1(\eta ,\xi;q),\varphi_2(\eta ,\xi;q),\varphi_3(\eta ,\xi;q)]& = \mathcal{M}[\varphi_3(\eta ,\xi;q)]+{s}\left(\frac{7}{8}-2 \tanh ^2(\eta )\right)\\&+\frac{1}{s^{\sigma}}\mathcal{M}\Big[6\varphi_3(\eta ,\xi;q)\frac{\partial \varphi_3(\eta ,\xi;q)}{\partial \eta }+\frac{\partial ^3\varphi_3(\eta ,\xi;q)}{\partial \eta^3}\\&-2\varphi_1(\eta ,\xi;q)\frac{\partial \varphi_1(\eta ,\xi;q)}{\partial \eta}-2\varphi_2(\eta ,\xi;q)\frac{\partial \varphi_2(\eta ,\xi;q)}{\partial \eta }\Big]. \end{split} \end{equation}$

(4.4)

Also, the operators of the Mohand are given as:

$\begin{equation} \begin{split} &\mathcal{M}[{{\mathbb{U}}_{{m}}}(\eta,\xi)-k_{{m}}{{\mathbb{U}}_{{m}-1}}(\eta,\xi)] = \hbar\mathfrak{h}(\eta,\xi) {\mathcal{R}}_{{1,m}}[\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}],\\ &\mathcal{M}[{{\mathbb{V}}_{{m}}}(\eta,\xi)-k_{{m}}{{\mathbb{V}}_{{m}-1}}(\eta,\xi)] = \hbar\mathfrak{h}(\eta,\xi) {\mathcal{R}}_{{2,m}}[\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}],\\ &\mathcal{M}[{{\mathbb{W}}_{{m}}}(\eta,\xi)-k_{{m}}{{\mathbb{W}}_{{m}-1}}(\eta,\xi)] = \hbar\mathfrak{h}(\eta,\xi) {\mathcal{R}}_{{3,m}}[\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}]. \end{split} \end{equation}$

(4.5)

Here,

$\begin{equation} \begin{split} {\mathcal{R}}_{{1,m}}[\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}]& = \mathcal{M}[{\mathbb{U}_{m-1}}(\eta ,\xi )]+{s}\left(1-\frac{k_m}{n}\right)\left(\cos (\eta ) \tanh (\eta )\right)\\ &+\frac{1}{s^{\sigma}}\mathcal{M}\Big[-\frac{\partial ^2\mathbb{V}_{m-1}(\eta ,\xi )}{\partial \eta^2}-\sum\limits_{i}^{m-1}\mathbb{V}_{i}(\eta ,\xi )\mathbb{W}_{m-1-i}(\eta ,\xi )\Big],\\ {\mathcal{R}}_{{2,m}}[\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}]& = \mathcal{M}[{\mathbb{V}_{m-1}}(\eta ,\xi )]+{s}\left(1-\frac{k_m}{n}\right)\left(\sin (\eta ) \tanh (\eta )\right)\\&+\frac{1}{s^{\sigma}}\mathcal{M}\Big[\frac{\partial ^2\mathbb{U}_{m-1}(\eta ,\xi )}{\partial \eta^2}+\sum\limits_{i}^{m-1}\mathbb{U}_i(\eta ,\xi )\mathbb{W}_{m-1-i}(\eta ,\xi )\Big],\\ {\mathcal{R}}_{{3,m}}[\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}]& = \mathcal{M}[{\mathbb{W}_{m-1}}(\eta ,\xi )]+{s}\left(1-\frac{k_m}{n}\right)\left(\frac{7}{8}-2 \tanh ^2(\eta )\right)\\&+\frac{1}{s^{\sigma}}\mathcal{M}\Big[6\sum\limits_{i}^{m-1}\Big[\mathbb{W}_{i}(\eta ,\xi )\times\frac{\partial \mathbb{W}_{m-1-i}(\eta ,\xi )}{\partial \eta }\Big]+\frac{\partial ^3\mathbb{W}_{m-1}(\eta ,\xi )}{\partial \eta^3}\\&-2\sum\limits_{i}^{m-1}\mathbb{U}_i(\eta ,\xi )\frac{\partial \mathbb{U}_{m-1-i}(\eta ,\xi )}{\partial \eta}-2\mathbb{V}_i(\eta ,\xi )\frac{\partial \mathbb{V}_{m-1-i}(\eta ,\xi )}{\partial \eta }\Big], \end{split} \end{equation}$

(4.6)

$\begin{equation} \begin{split} &{\mathbb{U}_{{m}}}(\eta ,\xi) = k_{{m}}{\mathbb{U}_{{m}-1}}(\eta ,\xi)+\hbar\mathcal{M}^{-1}[\mathfrak{h}(\eta ,\xi) {\mathcal{R}}_{{1,m}}(\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}),\\ &{\mathbb{V}_{{m}}}(\eta ,\xi) = k_{{m}}{\mathbb{V}_{{m}-1}}(\eta ,\xi)+\hbar\mathcal{M}^{-1}[\mathfrak{h}(\eta ,\xi) {\mathcal{R}}_{{2,m}}(\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}),\\ &{\mathbb{W}_{{m}}}(\eta ,\xi) = k_{{m}}{\mathbb{W}_{{m}-1}}(\eta ,\xi)+\hbar\mathcal{M}^{-1}[\mathfrak{h}(\eta ,\xi) {\mathcal{R}}_{{3,m}}(\overrightarrow{{\mathbb{U}}}_{{m}-1},\overrightarrow{{\mathbb{V}}}_{{m}-1},\overrightarrow{{\mathbb{W}}}_{{m}-1}). \end{split} \end{equation}$

(4.7)

The initial condition in Eq (4.7) allows us to derive the following:

$\begin{equation} \begin{split} &{\mathbb{U}}_{1}(\eta,\xi) = -\frac{\hbar \xi^{\sigma } \left(16 \cos (\eta) \text{sech}^2(\eta)-17 \sin (\eta) \tanh (\eta)\right)}{8 \Gamma (\sigma +1)},\\ &{\mathbb{V}}_{1}(\eta,\xi) = -\frac{\hbar \xi^{\sigma } \left(17 \cos (\eta) \tanh (\eta)+16 \sin (\eta) \text{sech}^2(\eta)\right)}{8 \Gamma (\sigma +1)},\\ &{\mathbb{W}}_{1}(\eta,\xi) = \frac{9 \hbar \xi^{\sigma } \tanh (\eta) \text{sech}^2(\eta)}{\Gamma (\sigma +1)}, \end{split} \end{equation}$

(4.8)

$\begin{equation} \begin{split} {\mathbb{U}}_{2}(\eta,\xi)& = \frac{\hbar \xi^{\sigma }}{256}\Big(\frac{32 (n+\hbar ) \left(17 \sin (\eta) \tanh (\eta)-16 \cos (\eta) \text{sech}^2(\eta)\right)}{\Gamma (\sigma +1)}\\& -\frac{\hbar \xi^{\sigma } \text{sech}^3(\eta) (\cos (\eta) (2337 \sinh (\eta)+289 \sinh (3 \eta))+64 \sin (\eta) (19 \cosh (2 \eta)+15) \text{sech}(\eta))}{\Gamma (2 \sigma +1)}\Big),\\ {\mathbb{V}}_{2}(\eta,\xi)& = \frac{\hbar \xi^{\sigma }}{256}\Big(-\frac{32 (n+\hbar ) \left(17 \cos (\eta) \tanh (\eta)+16 \sin (\eta) \text{sech}^2(\eta)\right)}{\Gamma (\sigma +1)}\\&+\frac{\hbar \xi^{\sigma } \text{sech}^3(\eta) (64 \cos (\eta) (19 \cosh (2 \eta)+15) \text{sech}(\eta)-2 \sin (\eta) \sinh (\eta) (289 \cosh (2 \eta)+1313))}{\Gamma (2 \sigma +1)}\Big),\\ {\mathbb{W}}_{2}(\eta,\xi)& = \frac{1}{4} \hbar \xi^{\sigma } \text{sech}^4(\eta) \left(\frac{18 (n+\hbar ) \sinh (2 \eta)}{\Gamma (\sigma +1)}+\frac{83 \hbar \xi^{\sigma } (\cosh (2 \eta)-2)}{\Gamma (2 \sigma +1)}\right), \end{split} \end{equation}$

(4.9)

and so on.

Additionally, the other components may be identified in this manner. The q-HATM solution of Eq (4.1) is derived as follows:

$\begin{equation} \begin{split} &{\mathbb{U}}(\eta,\xi) = \mathbb{U}_{0}+\sum\limits_{{m} = 1}^{\infty}{\mathbb{U}}_{{m}}\left(\frac{1}{n}\right)^{{m}},\\ &{\mathbb{V}}(\eta,\xi) = \mathbb{V}_{0}+\sum\limits_{{m} = 1}^{\infty}{\mathbb{V}}_{{m}}\left(\frac{1}{n}\right)^{{m}},\\ &{\mathbb{W}}(\eta,\xi) = \mathbb{W}_{0}+\sum\limits_{{m} = 1}^{\infty}{\mathbb{W}}_{{m}}\left(\frac{1}{n}\right)^{{m}}. \end{split} \end{equation}$

(4.10)

For $\sigma = 1$ , $\hbar = -1$ , and $n = 1$ , solutions $\sum_{m = 1}^{N}{\mathbb{U}}_{{m}}\left(\frac{1}{n}\right)^{{m}}$ , $\sum_{m = 1}^{N}{\mathbb{V}}_{{m}}\left(\frac{1}{n}\right)^{{m}}$ , and $\sum_{m = 1}^{N}{\mathbb{W}}_{{m}}\left(\frac{1}{n}\right)^{{m}}$ converge to the exact solutions as $N\rightarrow \infty$ .

$\begin{equation} \begin{aligned} {\mathbb{U}}(\eta,\xi)& = \cos (\eta ) \tanh (\eta )-\frac{\hbar \xi^{\sigma } \left(16 \cos (\eta) \text{sech}^2(\eta)-17 \sin (\eta) \tanh (\eta)\right)}{8 \Gamma (\sigma +1)}\nonumber\\ &+\frac{\hbar \xi^{\sigma }}{256}\Big(\frac{32 (n+\hbar ) \left(17 \sin (\eta) \tanh (\eta)-16 \cos (\eta) \text{sech}^2(\eta)\right)}{\Gamma (\sigma +1)}\nonumber\\ & -\frac{\hbar \xi^{\sigma } \text{sech}^3(\eta) (\cos (\eta) (2337 \sinh (\eta)+289 \sinh (3 \eta))+64 \sin (\eta) (19 \cosh (2 \eta)+15) \text{sech}(\eta))}{\Gamma (2 \sigma +1)}\Big)\nonumber\\ &+\cdots,\nonumber \end{aligned} \end{equation}$

$\begin{align} {\mathbb{V}}(\eta,\xi)& = \sin (\eta ) \tanh (\eta )-\frac{\hbar \xi^{\sigma } \left(17 \cos (\eta) \tanh (\eta)+16 \sin (\eta) \text{sech}^2(\eta)\right)}{8 \Gamma (\sigma +1)}\\ &+\frac{\hbar \xi^{\sigma }}{256}\Big(-\frac{32 (n+\hbar ) \left(17 \cos (\eta) \tanh (\eta)+16 \sin (\eta) \text{sech}^2(\eta)\right)}{\Gamma (\sigma +1)}\\ &+\frac{\hbar \xi^{\sigma } \text{sech}^3(\eta) (64 \cos (\eta) (19 \cosh (2 \eta)+15) \text{sech}(\eta)-2 \sin (\eta) \sinh (\eta) (289 \cosh (2 \eta)+1313))}{\Gamma (2 \sigma +1)}\Big)\\ &+\cdots,\\ {\mathbb{W}}(\eta,\xi)& = \frac{7}{8}-2 \tanh ^2(\eta )+\frac{9 \hbar \xi^{\sigma } \tanh (\eta) \text{sech}^2(\eta)}{\Gamma (\sigma +1)}+\frac{1}{4} \hbar \xi^{\sigma } \text{sech}^4(\eta) \Big(\frac{18 (n+\hbar ) \sinh (2 \eta)}{\Gamma (\sigma +1)}\\ &+\frac{83 \hbar \xi^{\sigma } (\cosh (2 \eta)-2)}{\Gamma (2 \sigma +1)}\Big)\\ &+\cdots. \end{align}$

(4.11)

4.2. The implementation of VITM

Consider the time fractional order system of Schrödinger-KdV equations:

$\begin{equation} \begin{split} D_{{{\xi}}}^{\sigma}\mathbb{U}(\eta ,\xi )& = \frac{\partial ^2\mathbb{V}(\eta ,\xi )}{\partial \eta^2}+\mathbb{V}(\eta ,\xi )\mathbb{W}(\eta ,\xi ),\\ D_{{{\xi}}}^{\sigma}\mathbb{V}(\eta ,\xi )& = -\frac{\partial ^2\mathbb{U}(\eta ,\xi )}{\partial \eta^2}-\mathbb{U}(\eta ,\xi )\mathbb{W}(\eta ,\xi ),\\ D_{{{\xi}}}^{\sigma}\mathbb{W}(\eta ,\xi )& = -6\mathbb{W}(\eta ,\xi )\frac{\partial \mathbb{W}(\eta ,\xi )}{\partial \eta }-\frac{\partial ^3\mathbb{W}(\eta ,\xi )}{\partial \eta^3}+2\mathbb{U}(\eta ,\xi )\frac{\partial \mathbb{U}(\eta ,\xi )}{\partial \eta}\\ &+2\mathbb{V}(\eta ,\xi )\frac{\partial \mathbb{V}(\eta ,\xi )}{\partial \eta }, \end{split} \end{equation}$

(4.12)

where $0 < \sigma \leq 1$ .

Initial conditions are given as:

(4.13)

By employing the recursive formula defined by Eq (3.21), we can get

$\begin{align} {\mathbb{U}}_{n+1}(\eta ,\xi)& = {\mathbb{U}_{n}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{U}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[\frac{\partial ^2\mathbb{V}_{n}(\eta ,\xi )}{\partial \eta^2}+\mathbb{V}_{n}(\eta ,\xi )\mathbb{W}_{n}(\eta ,\xi )]\Big], \\ {\mathbb{V}}_{n+1}(\eta ,\xi)& = {\mathbb{V}_{n}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{V}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[-\frac{\partial ^2\mathbb{U}_{n}(\eta ,\xi )}{\partial \eta^2}-\mathbb{U}_{n}(\eta ,\xi )\mathbb{W}_{n}(\eta ,\xi )]\Big],\\ {\mathbb{W}}_{n+1}(\eta ,\xi)& = {\mathbb{W}_{n}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{W}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[-6\mathbb{W}_{n}(\eta ,\xi )\frac{\partial \mathbb{W}_{n}(\eta ,\xi )}{\partial \eta }-\frac{\partial ^3\mathbb{W}_{n}(\eta ,\xi )}{\partial \eta^3}\\ &+2\mathbb{U}_{n}(\eta ,\xi )\frac{\partial \mathbb{U}_{n}(\eta ,\xi )}{\partial \eta}+2\mathbb{V}_{n}(\eta ,\xi )\frac{\partial \mathbb{V}_{n}(\eta ,\xi )}{\partial \eta }]\Big]. \end{align}$

(4.14)

The second approximation may be produced by inserting $n = 0$ , as shown in the expression below:

$\begin{equation} \begin{split} {\mathbb{U}}_{1}(\eta ,\xi)& = {\mathbb{U}_{0}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{U}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[\frac{\partial ^2\mathbb{V}_{0}(\eta ,\xi )}{\partial \eta^2}+\mathbb{V}_{0}(\eta ,\xi )\mathbb{W}_{0}(\eta ,\xi )]\Big],\\ {\mathbb{V}}_{1}(\eta ,\xi)& = {\mathbb{V}_{0}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{V}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[-\frac{\partial ^2\mathbb{U}_{0}(\eta ,\xi )}{\partial \eta^2}-\mathbb{U}_{0}(\eta ,\xi )\mathbb{W}_{0}(\eta ,\xi )]\Big],\\ {\mathbb{W}}_{1}(\eta ,\xi)& = {\mathbb{W}_{0}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{W}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[-6\mathbb{W}_{0}(\eta ,\xi )\frac{\partial \mathbb{W}_{0}(\eta ,\xi )}{\partial \eta }-\frac{\partial ^3\mathbb{W}_{0}(\eta ,\xi )}{\partial \eta^3}\\&+2\mathbb{U}_{0}(\eta ,\xi )\frac{\partial \mathbb{U}_{0}(\eta ,\xi )}{\partial \eta}+2\mathbb{V}_{0}(\eta ,\xi )\frac{\partial \mathbb{V}_{0}(\eta ,\xi )}{\partial \eta }]\Big]. \end{split} \end{equation}$

(4.15)

By simplification, we get

$\begin{equation} \begin{split} &{\mathbb{U}}_{1}(\eta ,\xi) = \frac{\xi ^{\sigma} \left(16 \cos (\eta) \text{sech}^2(\eta)-17 \sin (\eta) \tanh (\eta)\right)}{8 \Gamma (\sigma +1)}+\cos (\eta) \tanh (\eta),\\ &{\mathbb{V}}_{1}(\eta ,\xi) = \frac{\xi^{\sigma} \left(17 \cos (\eta) \tanh (\eta)+16 \sin (\eta) \text{sech}^2(\eta)\right)}{8 \Gamma (\sigma +1)}+\sin (\eta) \tanh (\eta),\\ &{\mathbb{W}}_{1}(\eta ,\xi) = -\frac{9 \xi ^{\sigma} \tanh (\eta) \text{sech}^2(\eta)}{\Gamma (\sigma +1)}+2 \text{sech}^2(\eta)-\frac{9}{8}. \end{split} \end{equation}$

(4.16)

By taking $n = 1$ , we have

$\begin{equation} \begin{split} {\mathbb{U}}_{2}(\eta ,\xi)& = {\mathbb{U}_{1}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{U}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[\frac{\partial ^2\mathbb{V}_{1}(\eta ,\xi )}{\partial \eta^2}+\mathbb{V}_{1}(\eta ,\xi )\mathbb{W}_{1}(\eta ,\xi )]\Big],\\ {\mathbb{V}}_{2}(\eta ,\xi)& = {\mathbb{V}_{1}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{V}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[-\frac{\partial ^2\mathbb{U}_{1}(\eta ,\xi )}{\partial \eta^2}-\mathbb{U}_{1}(\eta ,\xi )\mathbb{W}_{1}(\eta ,\xi )]\Big],\\ {\mathbb{W}}_{2}(\eta ,\xi)& = {\mathbb{W}_{1}}(\eta ,\xi) +\mathcal{M}^{-1}\Big[\frac{1}{{s}^{\sigma}}\sum\limits_{{k} = 0}^{m-1}{s}^{\sigma-{k}-1}\frac{\partial^{{k}}{\mathbb{W}} (\eta,0)}{\partial\xi^{{k}}}+\mathcal{M}[-6\mathbb{W}_{1}(\eta ,\xi )\frac{\partial \mathbb{W}_{1}(\eta ,\xi )}{\partial \eta }-\frac{\partial ^3\mathbb{W}_{1}(\eta ,\xi )}{\partial \eta^3}\\&+2\mathbb{U}_{1}(\eta ,\xi )\frac{\partial \mathbb{U}_{1}(\eta ,\xi )}{\partial \eta}+2\mathbb{V}_{1}(\eta ,\xi )\frac{\partial \mathbb{V}_{1}(\eta ,\xi )}{\partial \eta }]\Big]. \end{split} \end{equation}$

(4.17)

As a result of simplifying the expression, we arrive at the following final result:

$\begin{align} {\mathbb{U}}_{2}(\eta ,\xi)& = \cos (\eta) \tanh (\eta)+\frac{\xi^{\sigma} \left(16 \cos (\eta) \text{sech}^2(\eta)-17 \sin (\eta) \tanh (\eta)\right)}{8 \Gamma (\sigma +1)}\\ &-\frac{9 \Gamma (2 \sigma +1) \xi^{3 \sigma} \tanh (\eta) \text{sech}^2(\eta) \left(17 \cos (\eta) \tanh (\eta)+16 \sin (\eta) \text{sech}^2(\eta)\right)}{8 \Gamma (\sigma +1)^2 \Gamma (3 \sigma +1)}\\ &-\frac{\xi^{2 \sigma } \left(289 \cos (\eta) \tanh (\eta)+32 \text{sech}^2(\eta) (17 \sin (\eta)+2 \tanh (\eta) (8 \cos (\eta)+\sin (\eta) \tanh (\eta)))\right)}{64 \Gamma (2 \sigma +1)},\\ {\mathbb{V}}_{2}(\eta ,\xi)& = \frac{1}{64}\Big(64 \sin (\eta) \tanh (\eta)+\frac{8 \xi^{\sigma } \left(17 \cos (\eta) \tanh (\eta)+16 \sin (\eta) \text{sech}^2(\eta)\right)}{\Gamma (\sigma +1)}\\ &+\frac{72 \Gamma (2 \sigma +1) t^{3 \sigma } \tanh (\eta) \text{sech}^2(\eta) \left(16 \cos (\eta) \text{sech}^2(\eta)-17 \sin (\eta) \tanh (\eta)\right)}{\Gamma (\sigma +1)^2 \Gamma (3 \sigma +1)}\\ &+\frac{\xi^{2 \sigma } \left(32 \text{sech}^2(\eta) (17 \cos (\eta)+2 \tanh (\eta) (\cos (\eta) \tanh (\eta)-8 \sin (\eta)))-289 \sin (\eta) \tanh (\eta)\right)}{\Gamma (2 \sigma +1)}\Big),\\ {\mathbb{W}}_{2}(\eta ,\xi)& = \frac{\text{sech}^2(\eta)}{512}\Big(\frac{10624 \xi^{2 \sigma } (\cosh (2 \eta)-2) \text{sech}^2(\eta)}{\Gamma (2 \sigma +1)}\\ &+32 \left(-\frac{144 \xi^{\sigma } \tanh (\eta)}{\Gamma (\sigma +1)}-9 \cosh (2 \eta)+23\right)\\ &+\frac{\Gamma (2 \sigma +1) \xi^{3 \sigma } (-623550 \sinh (\eta)+123235 \sinh (3 \eta)+289 \sinh (5 \eta)) \text{sech}^5(\eta)}{\Gamma (\sigma +1)^2 \Gamma (3 \sigma +1)}\Big). \end{align}$

(4.18)

5. Graphical and tables discussion

The research evaluates the graphical impact of fractional ordering effects on Schrödinger and KdV equation solutions by utilizing both VITM and q-HATM. represents how the solution ${\mathbb{U}}(\eta, \xi)$ reacts when fractional orders $\sigma$ increase. displays the solutions with corresponding fractional orders of 0.4, 0.6, and 1.0. Profile diagram analysis establishes that increasing the $\sigma$ value leads to significant changes in solution forms, which proves how fractional order fundamentally affects system dynamic behavior. shows solutions that illustrate transformations in system behavior across variable $\sigma$ values. The solutions of ${\mathbb{U}}(\eta, \xi)$ generated using q-HATM and VITM methods appear in . Examining the precision details of q-HATM and VITM through method comparison aids researchers to determine their applicability for fractional differential equation solutions. The analysis of ${\mathbb{U}}(\eta, \xi)$ for $\xi = 0.01$ appears in through q-HATM and VITM evaluation methods. The presented tabulated information permits an immediate comparison of accuracy and operational performance characteristics. demonstrates how the methods maintain their accuracy level in ${\mathbb{U}}(\eta, \xi)$ calculations when operating at $\xi = 0.01$ . The limited error ranges strengthen the reliability of both q-HATM and VITM for their capability to approximate solutions for fractional differential equations. presents the analysis of solution ${\mathbb{U}}(\eta, \xi)$ while varying fractional order parameters just like did. A unified comparison appears in while correspond respectively to the values $\sigma = 0.4$ , $\sigma = 0.6$ , and $\sigma = 1.0$ . The systematic variations confirm the high sensitivity of ${\mathbb{V}}(\eta, \xi)$ to modifications in $\sigma$ because FC impacts wave dynamics thoroughly. Such a figure depicts the q-HATM and VITM generated solutions of ${\mathbb{V}}(\eta, \xi)$ to demonstrate their match and mismatch points. Analytical methods in FC require such comparative assessments to validate their robustness. A thorough analysis of ${\mathbb{V}}(\eta, \xi)$ at the specific condition of $\xi = 0.01$ is presented in to display the results obtained from both methods to evaluate their performance characteristics. demonstrates the accuracy and reliability of employed methods to track fractional-order systems through absolute errors presented for ${\mathbb{V}}(\eta, \xi)$ evaluated at $\xi = 0.01$ . represents an examination of ${\mathbb{W}}(\eta, \xi)$ solutions by varying fractional order values. displays the solutions of the system when parameter $\sigma$ equals 0.4, 0.6, and 1.0. presents an overall comparison of the solutions. The graphical representations establish that the fractional parameter function is a primary influence that directs the solution's configuration. compares q-HATM and VITM solutions for ${\mathbb{W}}(\eta, \xi)$ , allowing assessment of method performance capabilities and constraints when used in fractional differential equations. showcases an extensive review of the ${\mathbb{W}}(\eta, \xi)$ solution at $\xi = 0.01$ through both methods to evaluate their operational capabilities and computational resource consumption. focuses on the absolute errors of ${\mathbb{W}}(\eta, \xi)$ , which strengthens the evidence regarding q-HATM and VITM as methods that produce accurate solutions for complex fractional systems with $\xi = 0.01$ . The tabular and graphical assessments demonstrate that q-HATM and VITM succeed efficiently in resolving fractional-order Schrödinger and KdV equations. These visualizations verify the fractional parameter sensitivities and show the accuracy and convergence of the proposed approaches that significantly benefit FC applications in mathematical physics and engineering.

Figure 1. Analysis of fractional-order and its effect on the approximate solution (a)

$\sigma = 0.4$ , (b)

$\sigma = 0.6$ , (c)

$\sigma = 1.0$ , and (d) comparison of various values of

$\sigma$ of the solution

${\mathbb{U}}(\eta, \xi)$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. q-HMTM solution and MVIM solution comparison.

DownLoad: Full-Size Img PowerPoint

Table 1. Analysis of both methods for

$\mathbb{U}(\eta, \xi)$ at

$\xi = 0.01$ .

	$\mathbb{U}(\eta, \xi)$ for $\sigma=0.4$		$\mathbb{U}(\eta, \xi)$ for $\sigma=0.6$		$\mathbb{U}(\eta, \xi)$ for $\sigma=0.8$		$\mathbb{U}(\eta, \xi)$ for $\sigma=1.0$
$\eta$	q-HATM	VITM	q-HATM	VITM	q-HATM	VITM	q-HATM	VITM
0.5	0.36120935191	0.33861548115	0.44206697456	0.44097410128	0.42669166423	0.42664535753	0.41425805108	0.41425628931
1.0	0.07535823815	0.05590024405	0.32406890568	0.32312771582	0.38415640275	0.38411652310	0.40208860319	0.40208708594
1.5	-0.32972156568	-0.33284644536	-0.07709961858	-0.07725077008	0.01206320878	0.01205680426	0.04499401392	0.04499377026
2.0	-0.70578313652	-0.70388592079	-0.53169156256	-0.53159979358	-0.45236787232	-0.45236398393	-0.42032091350	-0.42032076556
2.5	-0.92543034468	-0.92376704004	-0.86907219111	-0.86899173649	-0.82382524949	-0.82382184051	-0.80321379853	-0.80321366884
3.0	-0.92017861851	-0.91938901088	-0.99124148703	-0.99120329343	-0.99167943639	-0.99167781807	-0.98805041295	-0.98805035138
3.5	-0.68816113531	-0.68788304269	-0.86729664860	-0.86728319717	-0.91298724593	-0.91298667597	-0.92716497971	-0.92716495802
4.0	-0.28644334453	-0.28637167633	-0.52912363819	-0.52912017157	-0.60860010156	-0.60859995468	-0.63700317136	-0.63700316578
4.5	0.18599199004	0.18600050937	-0.06065114716	-0.06065073508	-0.15431859025	-0.15431857279	-0.18993061663	-0.18993061597
5.0	0.61310384294	0.61309963502	0.42290082902	0.42290062548	0.33802542677	0.33802541815	0.30394877330	0.30394877298

| Show Table

DownLoad: CSV

Table 2. Comparison of absolute error of

$\mathbb{U}(\eta, \xi)$ at

$\xi = 0.01$ .

$\eta$	\| $q-HATM-VITM\|_{\sigma=0.4}$	\| $q-HATM-VITM\|_{\sigma=0.6}$	\| $q-HATM-VITM\|_{\sigma=0.8}$	\| $q-HATM-VITM\|_{\sigma=1.0}$
0.5	2.25939 $\times 10^{-2}$	1.09287 $\times 10^{-3}$	4.63067 $\times 10^{-5}$	1.76177 $\times 10^{-6}$
1.0	1.94580 $\times 10^{-2}$	9.41190 $\times 10^{-4}$	3.98796 $\times 10^{-5}$	1.51725 $\times 10^{-6}$
1.5	3.12488 $\times 10^{-3}$	1.51152 $\times 10^{-4}$	6.40452 $\times 10^{-6}$	2.43665 $\times 10^{-7}$
2.0	1.89722 $\times 10^{-3}$	9.17690 $\times 10^{-5}$	3.88839 $\times 10^{-6}$	1.47937 $\times 10^{-7}$
2.5	1.66330 $\times 10^{-3}$	8.04546 $\times 10^{-5}$	3.40898 $\times 10^{-6}$	1.29697 $\times 10^{-7}$
3.0	7.89608 $\times 10^{-4}$	3.81936 $\times 10^{-5}$	1.61832 $\times 10^{-6}$	6.15702 $\times 10^{-8}$
3.5	2.78093 $\times 10^{-4}$	1.34514 $\times 10^{-5}$	5.69958 $\times 10^{-7}$	2.16845 $\times 10^{-8}$
4.0	7.16682 $\times 10^{-5}$	3.46662 $\times 10^{-6}$	1.46886 $\times 10^{-7}$	5.58838 $\times 10^{-9}$
4.5	8.51933 $\times 10^{-6}$	4.12083 $\times 10^{-7}$	1.74606 $\times 10^{-8}$	6.64301 $\times 10^{-10}$
5.0	4.20793 $\times 10^{-6}$	2.03539 $\times 10^{-7}$	8.62425 $\times 10^{-9}$	3.28116 $\times 10^{-10}$

| Show Table

DownLoad: CSV

Figure 3. Analysis of fractional-order and its effect on the approximate solution (a)

$\sigma = 0.4$ , (b)

$\sigma = 0.6$ , (c)

$\sigma = 1.0$ , and (d) comparison of various values of

$\sigma$ of the solution

${\mathbb{V}}(\eta, \xi)$ .

DownLoad: Full-Size Img PowerPoint

Table 3. Analysis of both methods for

$\mathbb{V}(\eta, \xi)$ at

$\xi = 0.01$ .

	$\mathbb{V}(\eta, \xi)$ for $\sigma=0.4$		$\mathbb{V}(\eta, \xi)$ for $\sigma=0.6$		$\mathbb{V}(\eta, \xi)$ for $\sigma=0.8$		$\mathbb{V}(\eta, \xi)$ for $\sigma=1.0$
$\eta$	q-HATM	VITM	q-HATM	VITM	q-HATM	VITM	q-HATM	VITM
0.5	0.60780518968	0.62052294719	0.34873394344	0.34934910574	0.26672730729	0.26675337266	0.23789045679	0.23789144847
1.0	0.84275800769	0.83158438291	0.74172535549	0.74118488342	0.68218555413	0.68216265350	0.65652216132	0.65652129005
1.5	0.84963577507	0.83772282643	0.91892881087	0.91835257745	0.91395006747	0.91392565158	0.90758335221	0.90758242329
2.0	0.61966595118	0.61463070669	0.80836697215	0.80812341564	0.85497350506	0.85496318520	0.86911108695	0.86911069433
2.5	0.21537795275	0.21406841731	0.46328353857	0.46322019589	0.54469938306	0.54469669913	0.57383336205	0.57383325994
3.0	-0.25290584226	-0.25302592632	-0.00986821399	-0.00987402250	0.08371570260	0.08371545649	0.11947978154	0.11947977217
3.5	-0.66337152497	-0.66326844271	-0.48495767029	-0.48495268416	-0.40309544956	-0.40309523829	-0.36995510627	-0.36995509823
4.0	-0.91250404947	-0.91242128204	-0.84211744513	-0.84211344164	-0.79227766347	-0.79227749384	-0.77002506874	-0.77002506229
4.5	-0.93836059593	-0.93832117785	-0.99302480260	-0.99302289593	-0.98744366518	-0.98744358439	-0.98155603065	-0.98155602757
5.0	-0.73441447988	-0.73440024598	-0.90064889998	-0.90064821148	-0.94067965898	-0.94067962980	-0.95259683520	-0.95259683409

| Show Table

DownLoad: CSV

Table 4. Comparison of absolute error of

$\mathbb{V}(\eta, \xi)$ at

$\xi = 0.01$ .

$\eta$	\| $q-HATM-VITM\|_{\sigma=0.4}$	\| $q-HATM-VITM\|_{\sigma=0.6}$	\| $q-HATM-VITM\|_{\sigma=0.8}$	\| $q-HATM-VITM\|_{\sigma=1.0}$
0.5	1.27178 $\times 10^{-2}$	6.15162 $\times 10^{-4}$	2.60654 $\times 10^{-5}$	9.91676 $\times 10^{-7}$
1.0	1.11736 $\times 10^{-2}$	5.40472 $\times 10^{-4}$	2.29006 $\times 10^{-5}$	8.71271 $\times 10^{-7}$
1.5	1.19129 $\times 10^{-2}$	5.76233 $\times 10^{-4}$	2.44159 $\times 10^{-5}$	9.28921 $\times 10^{-7}$
2.0	5.03524 $\times 10^{-3}$	2.43557 $\times 10^{-4}$	1.03199 $\times 10^{-5}$	3.92627 $\times 10^{-7}$
2.5	1.30954 $\times 10^{-3}$	6.33427 $\times 10^{-5}$	2.68393 $\times 10^{-6}$	1.02112 $\times 10^{-7}$
3.0	1.20084 $\times 10^{-4}$	5.80851 $\times 10^{-6}$	2.46115 $\times 10^{-7}$	9.36364 $\times 10^{-9}$
3.5	1.03082 $\times 10^{-4}$	4.98612 $\times 10^{-6}$	2.11270 $\times 10^{-7}$	8.03791 $\times 10^{-9}$
4.0	8.27674 $\times 10^{-5}$	4.00349 $\times 10^{-6}$	1.69634 $\times 10^{-7}$	6.45385 $\times 10^{-9}$
4.5	3.94181 $\times 10^{-5}$	1.90667 $\times 10^{-6}$	8.07884 $\times 10^{-8}$	3.07365 $\times 10^{-9}$
5.0	1.42339 $\times 10^{-5}$	6.88499 $\times 10^{-7}$	2.91727 $\times 10^{-8}$	1.10990 $\times 10^{-9}$

| Show Table

DownLoad: CSV

Figure 4. q-HMTM solution and MVIM solution comparison.

DownLoad: Full-Size Img PowerPoint

Figure 5. Analysis of fractional-order and its effect on the approximate solution (a)

$\sigma = 0.4$ , (b)

$\sigma = 0.6$ , (c)

$\sigma = 1.0$ , and (d) comparison of various values of

$\sigma$ of the solution

${\mathbb{W}}(\eta, \xi)$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. q-HMTM solution and MVIM solution comparison.

DownLoad: Full-Size Img PowerPoint

Table 5. Analysis of both methods for

$\mathbb{W}(\eta, \xi)$ at

$\xi = 0.01$ .

	$\mathbb{W}(\eta, \xi)$ for $\sigma=0.4$		$\mathbb{W}(\eta, \xi)$ for $\sigma=0.6$		$\mathbb{W}(\eta, \xi)$ for $\sigma=0.8$		$\mathbb{W}(\eta, \xi)$ for $\sigma=1.0$
$\eta$	q-HMTM	MVIM	q-HMTM	MVIM	q-HMTM	MVIM	q-HMTM	MVIM
0.5	-0.29452181924	-0.51342166394	0.19573362966	0.18514536901	0.35709374782	0.35664510709	0.41489347445	0.41487640557
1.0	-0.62532024144	-0.41562069162	-0.46502454994	-0.45488131071	-0.35984039106	-0.35941060656	-0.31351534831	-0.31349899683
1.5	-0.87911318876	-0.78529480710	-0.84778711073	-0.84324908340	-0.80087565389	-0.80068337076	-0.77803436109	-0.77802704554
2.0	-1.02249937462	-1.00232347897	-1.01751300066	-1.01653708566	-0.99907327449	-0.99903192349	-0.98969712693	-0.98969555370
2.5	-1.08541848196	-1.08166996825	-1.08466122354	-1.08447990665	-1.07771610556	-1.07770842288	-1.07412383315	-1.07412354085
3.0	-1.11017160848	-1.10940271801	-1.11004963325	-1.11001244175	-1.10747278314	-1.10747120728	-1.10613130885	-1.10613124890
3.5	-1.11950807646	-1.11931399352	-1.11948496307	-1.11947557521	-1.11853403114	-1.11853363336	-1.11803779954	-1.11803778440
4.0	-1.12297460963	-1.12291554550	-1.12296908001	-1.12296622306	-1.12261885089	-1.12261872984	-1.12243592637	-1.12243592177
4.5	-1.12425421872	-1.12423417578	-1.12425258832	-1.12425161883	-1.12412369198	-1.12412365090	-1.12405634751	-1.12405634595
5.0	-1.12472555004	-1.12471840557	-1.12472500497	-1.12472465939	-1.12467757932	-1.12467756467	-1.12465279786	-1.12465279730

| Show Table

DownLoad: CSV

Table 6. Comparison of absolute error of

$\mathbb{W}(\eta, \xi)$ at

$\xi = 0.01$ .

$\eta$	\| $q-HMTM-MVIM\|_{\sigma=0.4}$	\| $q-HMTM-MVIM\|_{\sigma=0.6}$	\| $q-HMTM-MVIM\|_{\sigma=0.8}$	\| $q-HMTM-MVIM\|_{\sigma=1.0}$
0.5	2.18900 $\times 10^{-1}$	1.05883 $\times 10^{-2}$	4.48641 $\times 10^{-4}$	1.70689 $\times 10^{-5}$
1.0	2.09700 $\times 10^{-1}$	1.01432 $\times 10^{-2}$	4.29784 $\times 10^{-4}$	1.63515 $\times 10^{-5}$
1.5	9.38184 $\times 10^{-2}$	4.53803 $\times 10^{-3}$	1.92283 $\times 10^{-4}$	7.31556 $\times 10^{-6}$
2.0	2.01759 $\times 10^{-2}$	9.75915 $\times 10^{-4}$	4.13510 $\times 10^{-5}$	1.57323 $\times 10^{-6}$
2.5	3.74851 $\times 10^{-3}$	1.81317 $\times 10^{-4}$	7.68267 $\times 10^{-6}$	2.92293 $\times 10^{-7}$
3.0	7.68890 $\times 10^{-4}$	3.71915 $\times 10^{-5}$	1.57586 $\times 10^{-6}$	5.99548 $\times 10^{-8}$
3.5	1.94083 $\times 10^{-4}$	9.38786 $\times 10^{-6}$	3.97778 $\times 10^{-7}$	1.51338 $\times 10^{-8}$
4.0	5.90641 $\times 10^{-5}$	2.85695 $\times 10^{-6}$	1.21053 $\times 10^{-7}$	4.60557 $\times 10^{-9}$
4.5	2.00429 $\times 10^{-5}$	9.69484 $\times 10^{-7}$	4.10785 $\times 10^{-8}$	1.56286 $\times 10^{-9}$
5.0	7.14447 $\times 10^{-6}$	3.45580 $\times 10^{-7}$	1.46428 $\times 10^{-8}$	5.57095 $\times 10^{-10}$

| Show Table

DownLoad: CSV

6. Conclusions

This research investigated the solution of Schrödinger and KdV equations using q-HATM and VITM under fractional-order differential equations. I changed nonlinear equations using Caputo fractional operator into efficient iterative algorithms for obtaining series solutions that showed rapid convergence. The two methods used for solving fractional differential equations showed ease of calculation while delivering precise results and confirmed the validity of obtained solutions through their established effectiveness. The utilized methods delivered powerful solutions for solving difficult wave oscillation and radiation problems. The research examined how fractional-order parameters shape nonlinear wave model dynamics, whereas FC proved essential for mathematical physics and engineering fields. The results highlighted the understanding of fractional-order parameters' influence on the dynamics of nonlinear wave models and deepened the appreciation of FC in mathematical physics and engineering. The proposed techniques that garnered attention from graphical illustrations increased confidence in numerically solving essential components of wave representation. These results can be further built upon by applying them to multidimensional and multi-order nonlinearity models, extending their reach across disciplines, including engineering and science.

Use of Generative-AI tools declaration

The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The author declares that he has no conflict of interest.

References

[1]	K. Huang, Statistical mechanics, New York: Wiley, 2008.
[2]	C. P. Bachas, Computer-intractability of the frustration model of a spin glass, J. Phys. A: Math. Gen., 17 (1984), L709. https://doi.org/10.1088/0305-4470/17/13/006 doi: 10.1088/0305-4470/17/13/006
[3]	F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. A: Math. Gen., 15 (1982), 3241. https://doi.org/10.1088/0305-4470/15/10/028 doi: 10.1088/0305-4470/15/10/028
[4]	S. F. Edwards, P. W. Anderson, Theory of spin glasses, J. Phys. F: Met. Phys., 5 (1975), 965–974. https://doi.org/10.1088/0305-4608/5/5/017 doi: 10.1088/0305-4608/5/5/017
[5]	S. Istrail, Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces (extended abstract), Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, 2000, 87–96. https://doi.org/10.1145/335305.335316 doi: 10.1145/335305.335316
[6]	D. L. Stein, C. M. Newman, Spin glasses and complexity, Princeton University Press, 2013.
[7]	E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Phys., 31 (1925), 253–258. https://doi.org/10.1007/BF02980577 doi: 10.1007/BF02980577
[8]	L. Onsager, Crystal statistics I: a two-dimensional model with an order-disorder transition, Phys. Rev., 65 (1944), 117–149. https://doi.org/10.1103/PhysRev.65.117 doi: 10.1103/PhysRev.65.117
[9]	Z. D. Zhang, Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices, Phil. Mag., 87 (2007), 5309–5419. https://doi.org/10.1080/14786430701646325 doi: 10.1080/14786430701646325
[10]	M. Garey, D. S. Johnson, Computers and intractability, A guide to the Theory of NP-completeness, W. H. Freeman and Company, San Francisco, 1979.
[11]	C. H. Papadimitriou, Computational complexity, Addison-Wesley, 1994.
[12]	S. Cook, The complexity of theorem-proving procedures, Proceedings of the Third Annual ACM Symposium on Theory of Computing, 1971,151–158. https://doi.org/10.1145/800157.805047 doi: 10.1145/800157.805047
[13]	L. A. Levin, Universal sequential search problems, Probl. Inform. Transm., 9 (1973), 265–266.
[14]	Z. D. Zhang, Computational complexity of spin-glass three-dimensional (3D) Ising model, J. Mater. Sci. Tech., 44 (2020), 116–120. https://doi.org/10.1016/j.jmst.2019.12.009 doi: 10.1016/j.jmst.2019.12.009
[15]	Z. D. Zhang, Mapping between spin-glass three-dimensional (3D) Ising model and Boolean satisfiability problem, Mathematics, 11 (2023), 237. https://doi.org/10.3390/math11010237 doi: 10.3390/math11010237
[16]	T. Dantzig, Numbers: the language of science, London: George Allen & Unwin, Ltd., 1930.
[17]	G. B. Mathews, On the partition of numbers, Proc. London Math. Soc., 28 (1897), 486–490. https://doi.org/10.1112/plms/s1-28.1.486 doi: 10.1112/plms/s1-28.1.486
[18]	S. Martello, D. Pisinger, P. Toth, New trends in exact algorithms for the 0-1 knapsack problem, Eur. J. Oper. Res., 123 (2000), 325–332. https://doi.org/10.1016/S0377-2217(99)00260-X doi: 10.1016/S0377-2217(99)00260-X
[19]	D. Pisinger, Where are the hard knapsack problems? Comput. Oper. Res., 32 (2005), 2271–2284. https://doi.org/10.1016/j.cor.2004.03.002
[20]	D. Fayard, G. Plateau, Resolution of the 0-1 knapsack problem: comparison of methods, Math. Program., 8 (1975), 272–307. https://doi.org/10.1007/BF01580448 doi: 10.1007/BF01580448
[21]	S. Sahni, Approximate algorithms for the 0-1 knapsack problem, J. ACM, 22 (1975), 115–124. https://doi.org/10.1145/321864.321873 doi: 10.1145/321864.321873
[22]	S. Al-Shihabi, A novel core-based optimization framework for binary integer programs- the multidemand multidimesional knapsack problem as a test problem, Oper. Res. Perspect., 8 (2021), 100182. https://doi.org/10.1016/j.orp.2021.100182 doi: 10.1016/j.orp.2021.100182
[23]	D. E. Armstrong, S. H. Jacobson, Data-independent neighborhood functions and strict local optima, Discrete Appl. Math., 146 (2005), 233–243. https://doi.org/10.1016/j.dam.2004.09.007 doi: 10.1016/j.dam.2004.09.007
[24]	D. S. Johnson, Approximation algorithms for combinatorial problems, J. Comput. Syst. Sci., 9 (1974), 256–278. https://doi.org/10.1016/S0022-0000(74)80044-9 doi: 10.1016/S0022-0000(74)80044-9
[25]	P. Toth, Optimization engineering techniques for the exact solution of NP-hard combinatorial optimization problems, Eur. J. Oper. Res., 125 (2000), 222–238. https://doi.org/10.1016/S0377-2217(99)00453-1 doi: 10.1016/S0377-2217(99)00453-1
[26]	G. Venkataraman, G. Athithan, Spin glass, the travelling salesman problem, neural networks and all that, Pramana-J. Phys., 36 (1991), 1–77. https://doi.org/10.1007/BF02846491 doi: 10.1007/BF02846491
[27]	Z. D. Zhang, O. Suzuki, N. H. March, Clifford algebra approach of 3D Ising model, Adv. Appl. Clifford Algebras, 29 (2019), 12. https://doi.org/10.1007/s00006-018-0923-2 doi: 10.1007/s00006-018-0923-2
[28]	O. Suzuki, Z. D. Zhang, A method of Riemann-Hilbert problem for Zhang's conjecture 1 in a ferromagnetic 3D Ising model: trivialization of topological structure, Mathematics, 9 (2021), 776. https://doi.org/10.3390/math9070776 doi: 10.3390/math9070776
[29]	B. C. Li, W. Wang, Exploration of dynamic phase transition of 3D Ising model with a new long-range interaction by using the Monte Carlo method, Chin. J. Phys., 90 (2024), 15–30. https://doi.org/10.1016/j.cjph.2024.05.021 doi: 10.1016/j.cjph.2024.05.021
[30]	K. Ghosh, C. J. Lobb, R. L. Greene, Critical phenomena in the double-exchange ferromagnet La_0.7Sr_0.3MnO₃, Phys. Rev. Lett., 81 (1998), 4740–4743. https://doi.org/10.1103/PhysRevLett.81.4740 doi: 10.1103/PhysRevLett.81.4740
[31]	J. T. Ho, J. D. Litster, Magnetic equation of state of CrBr₃ near the critical point, Phys. Rev. Lett., 22 (1969), 603–606. https://doi.org/10.1103/PhysRevLett.22.603 doi: 10.1103/PhysRevLett.22.603
[32]	Y. Liu, V. N. Ivanovski, C. Petrovic, Critical behavior of the van der Waals bonded ferromagnet Fe_3-xGeTe₂, Phys. Rev. B, 96 (2017), 144429. https://doi.org/10.1103/PhysRevB.96.144429 doi: 10.1103/PhysRevB.96.144429
[33]	F. X. Liu, L. M. Duan, Computational characteristics of the random-field Ising model with long-range interaction, Phys. Rev. A, 108 (2023), 012415. https://doi.org/10.1103/PhysRevA.108.012415 doi: 10.1103/PhysRevA.108.012415
[34]	S. Nagy, R. Paredes, J. M. Dudek, L. Dueñas-Osorio, M. Y. Vardi, Ising model partition-function computation as a weighted counting problem, Phys. Rev. E, 109 (2024), 055301. https://doi.org/10.1103/PhysRevE.109.055301 doi: 10.1103/PhysRevE.109.055301
[35]	H. J. Xu, S. Dasgupta, A. Pothen, A. Banerjee, Dynamic asset allocation with expected shortfall via quantum annealing, Entropy, 25 (2023), 541. https://doi.org/10.3390/e25030541 doi: 10.3390/e25030541
[36]	R. Ladner, On the structure of polynomial time reducibility, J. ACM, 22 (1975), 155–171. https://doi.org/10.1145/321864.321877 doi: 10.1145/321864.321877
[37]	P. Jonsson, V. Lagerkvist, G. Nordh, Constructing NP-intermediate problems by blowing holes with parameters of various properties, Theor. Comput. Sci., 581 (2015), 67–82. https://doi.org/10.1016/j.tcs.2015.03.009 doi: 10.1016/j.tcs.2015.03.009
[38]	R. Bellman, A Markovian decision process, J. Math. Mech., 6 (1957), 679–684. https://doi.org/10.1512/iumj.1957.6.56038 doi: 10.1512/iumj.1957.6.56038
[39]	O. Kyriienko, H. Sigurdsson, T. C. H. Liew, Probabilistic solving of NP-hard problems with bistable nonlinear optical networks, Phys. Rev. B, 99 (2019), 195301. https://doi.org/10.1103/PhysRevB.99.195301 doi: 10.1103/PhysRevB.99.195301
[40]	G. B. Dantzig, Discrete-variable extremum problems, Oper. Res., 5 (1957), 161–306. https://doi.org/10.1287/opre.5.2.266 doi: 10.1287/opre.5.2.266
[41]	R. Bellman, Letter to the Editor–Comment on Dantzig's paper on discrete-variable extremum problems, Oper. Res., 5 (1957), 613–738. https://doi.org/10.1287/opre.5.5.723 doi: 10.1287/opre.5.5.723
[42]	V. Cacchiani, M. Iori, A. Locatelli, S. Martello, Knapsack problems–an overview of recent advances. Part I: single knapsack problems, Comput. Oper. Res., 143 (2022), 105692. https://doi.org/10.1016/j.cor.2021.105692 doi: 10.1016/j.cor.2021.105692
[43]	V. Cacchiani, M. Iori, A. Locatelli, S. Martello, Knapsack problems–an overview of recent advances. Part Ⅱ: Multiple, multidimensional, and quadratic knapsack problems, Comput. Oper. Res., 143 (2022), 105693. https://doi.org/10.1016/j.cor.2021.105693
[44]	H. Kellerer, U. Pferschy, D. Pisinger, Knapsack problems, Springer-Verlag, 2004. https://doi.org/10.1007/978-3-540-24777-7
[45]	H. Nishimori, Statistical physics of spin glasses and information processing: an introduction, Oxford University Press, 2001.
[46]	S. Kirkpatrick, D. Sherrington, Infinite-ranged models of spin-glasses, Phys. Rev. B, 17 (1978), 4384–4403. https://doi.org/10.1103/PhysRevB.17.4384 doi: 10.1103/PhysRevB.17.4384
[47]	D. Sherrington, S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett., 35 (1975), 1792–1796. https://doi.org/10.1103/PhysRevLett.35.1792 doi: 10.1103/PhysRevLett.35.1792
[48]	S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. U. Hwang, Complex networks: structure and dynamics, Phys. Rep., 424 (2006), 175–308. https://doi.org/10.1016/j.physrep.2005.10.009 doi: 10.1016/j.physrep.2005.10.009
[49]	G. E. Hinton, R. R. Salakhutdinov, Reducing the dimensionality of data with neural networks, Science, 313 (2006), 504–507. https://doi.org/10.1126/science.1127647 doi: 10.1126/science.1127647
[50]	M. K. Kwan, Graphic programming using odd or even points, Chin. Math., 1(1962), 273–277.
[51]	U. U. Haus, K. Niermann, K. Truemper, R. Weismantel, Logic integer programming models for signaling networks, J. Comput. Biol., 16 (2009), 725–743. https://doi.org/10.1089/cmb.2008.0163 doi: 10.1089/cmb.2008.0163
[52]	D. Bertsimas, R. Demir, An approximate dynamic programming approach to multidimensional Knapsack problems, Manag. Sci., 48 (2002), 453–590. https://doi.org/10.1287/mnsc.48.4.550.208 doi: 10.1287/mnsc.48.4.550.208
[53]	G. J. Woeginger, When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? J. Comput., 12 (2000), 1–82. https://doi.org/10.1287/ijoc.12.1.57.11901
[54]	E. Balas, N. Simonetti, Linear time dynamic-programming algorithms for new classes of restricted TSPs: a computational study, J. Comput., 13 (2001), 1–94. https://doi.org/10.1287/ijoc.13.1.56.9748 doi: 10.1287/ijoc.13.1.56.9748
[55]	A. A. Melkman, T. Akutsu, An improved satisfiability algorithm for nested canalyzing functions and its application to determining a singleton attractor of a Boolean network, J. Comput. Biol., 20 (2013), 958–969. https://doi.org/10.1089/cmb.2013.0060 doi: 10.1089/cmb.2013.0060
[56]	P. C. Chu, J. E. Beasley, A genetic algorithm for the multidimensional Knapsack problem, J. Heuristics, 4 (1998), 63–86. https://doi.org/10.1023/A:1009642405419 doi: 10.1023/A:1009642405419
[57]	A. K. Hartmann, Ground-state behavior of the three-dimensional ±J random-bond Ising model, Phys. Rev. B, 59 (1999), 3617. https://doi.org/10.1103/PhysRevB.59.3617 doi: 10.1103/PhysRevB.59.3617
[58]	L. V. Snyder, M. S. Daskin, A random-key genetic algorithm for the generalized traveling salesman problem, Eur. J. Oper. Res., 174 (2006), 38–53. https://doi.org/10.1016/j.ejor.2004.09.057 doi: 10.1016/j.ejor.2004.09.057
[59]	P. Larranaga, C. M. H. Kuijpers, R. H. Murga, I. Inza, S. Dizdarevic, Genetic algorithms for the travelling salesman problem: a review of representations and operators, Artif. Intell. Rev., 13 (1999), 129–170. https://doi.org/10.1023/A:1006529012972 doi: 10.1023/A:1006529012972

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)