Applicability of NeQuick G ionospheric model for single-frequency GNSS users over India

K Siri Lakshatha; D Venkata Ratnam; K Sivakrishna; K Siri Lakshatha; D Venkata Ratnam; K Sivakrishna

doi:10.3934/geosci.2022008

AIMS Geosciences

2022, Volume 8, Issue 1: 127-136. doi: 10.3934/geosci.2022008

Previous Article Next Article

Research article Special Issues

Applicability of NeQuick G ionospheric model for single-frequency GNSS users over India

Koneru Lakshmaiah Education Foundation, Greenfields, Vaddeswaram, Guntur, Andhra Pradesh, India

Received: 01 November 2021 Revised: 09 December 2021 Accepted: 28 January 2022 Published: 09 February 2022

The major source of error in the positioning of GNSS is from the region of Ionosphere. The single-frequency GNSS receiver cannot eliminate the ionospheric error due to dispersive medium and frequency-dependent. The low-cost GNSS receivers are highly dependent on single-frequency approaches of Ionosphere region popularly known as Klobuchar, NeQuick G, and BDS2 methods to estimate the data of position, velocity and time. The regional satellite navigation system of India, known as Navigation with Indian Constellation (NavIC) adopted ionospheric models based on single-frequency namely, Klobuchar and grid-based correction models. The Klobuchar modelos accuracy is less for predicting ionospheric delays in low latitude regions like India under Equatorial Ionization Anomaly (EIA) conditions. In this paper, the NeQuick G modelos applicability for NavIC users over the Indian region is investigated. NeQuick G modelos performance is validated with dense GPS TEC network data of 26 stations spread across India and IRI-2016 model, during 2014, 2015 and 2016. The predicted TEC results indicate that EIA structures are well captured by NeQuick G and IRI-2016 models. The results indicate that both NeQuick G and IRI-2016 models well predict season asymmetry and decrease of TEC intensity due to descending phase solar cycle activity. It is found that NeQuick G is one of the contenders of single frequency ionospheric models for GNSS/NavIC users in India.

Keywords:

Citation: K Siri Lakshatha, D Venkata Ratnam, K Sivakrishna. Applicability of NeQuick G ionospheric model for single-frequency GNSS users over India[J]. AIMS Geosciences, 2022, 8(1): 127-136. doi: 10.3934/geosci.2022008

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]	Chekole DA, Giday NM, Nigussie M (2019) Performance of NeQuick-2, IRI-Plas 2017 and GIM models over Ethiopia during varying solar activity periods. J Atmos Sol-Terr Phys 195: 105117. https://doi.org/10.1016/j.jastp.2019.105117 doi: 10.1016/j.jastp.2019.105117
[2]	Panda SK, Gedam SS, Jin S (2015) Ionospheric TEC variations at low latitude Indian region. In Jin S, eds. Satellite Positioning-Methods, Models and Applications. Tech-Publisher, Rijeka, Croatia, 149–174. https://doi.org/10.5772/59988
[3]	Appleton EV (1946) Two anomalies in the ionosphere. Nature 157: 691. https://doi.org/10.1038/157691a0 doi: 10.1038/157691a0
[4]	Desai MV, Shah SN (2018) The GIVE ionospheric delay correction approach to improve positional accuracy of NavIC/IRNSS single-frequency receiver. Curr Sci 114: 1665–1676. https://doi.org/10.18520/cs/v114/i08/1665-1676 doi: 10.18520/cs/v114/i08/1665-1676
[5]	Gulyaeva TL, Huang X, Reinisch BW (2002) Plasmaspheric extension of topside electron density profiles. Adv Space Res 29: 825–831. https://doi.org/10.1016/S0273-1177(02)00038-8 doi: 10.1016/S0273-1177(02)00038-8
[6]	Gordiyenko GI, Maltseva OA, Arikan F, et al. (2018) The performance of the IRI-Plas model as compared with Alouette II and GIM-TEC data over the midlatitude station Alma-Ata. J Atmos Sol-Terr Phys 179: 504–516. https://doi.org/10.1016/j.jastp.2018.08.007 doi: 10.1016/j.jastp.2018.08.007
[7]	Zakharenkova IE, Cherniak IV, Krankowski A, et al. (2015) Vertical TEC representation by IRI 2012 and IRI Plas models for European midlatitudes. Adv Space Res 55: 2070–2076. https://doi.org/10.1016/j.asr.2014.07.027 doi: 10.1016/j.asr.2014.07.027
[8]	Adebiyi SJ, Adimula IA, Oladipo OA, et al. (2016) Assessment of IRI and IRI‐Plas models over the African equatorial and low‐latitude region. J Geophys Res Space Phys 121: 7287–7300. https://doi.org/10.1002/2016JA022697 doi: 10.1002/2016JA022697
[9]	Ezquer RG, Scidá LA, Orué YM, et al. (2018) NeQuick 2 and IRI Plas VTEC predictions for low latitude and South American sector. Adv Space Res 61: 1803–1818. https://doi.org/10.1016/j.asr.2017.10.003 doi: 10.1016/j.asr.2017.10.003
[10]	Atici R (2018) Comparison of GPS TEC with modelled values from IRI 2016 and IRI-PLAS over Istanbul, Turkey. Astrophys Space Sci 363: 231. https://doi.org/10.1007/s10509-018-3457-0 doi: 10.1007/s10509-018-3457-0
[11]	Filjar R, Weintrit A, Iliev TB, et al. (2020) Predictive Model of Total Electron Content during Moderately Disturbed Geomagnetic Conditions for GNSS Positioning Performance Improvement. In 2020 IEEE 23rd International Conference on Information Fusion (FUSION), 1–6. https://doi.org/10.23919/FUSION45008.2020.9190264
[12]	Mallika IL, Ratnam DV, Raman S, et al. (2020) A New Ionospheric Model for Single Frequency GNSS User Applications Using Klobuchar Model Driven by Auto Regressive Moving Average (SAKARMA) Method Over Indian Region. IEEE Access 8: 54535–54553. https://doi.org/10.1109/ACCESS.2020.2981365 doi: 10.1109/ACCESS.2020.2981365
[13]	Di Giovanni G, Radicella SM (1990) An analytical model of the electron density profile in the ionosphere. Adv Space Res 10: 27–30. https://doi.org/10.1016/0273-1177(90)90301-F doi: 10.1016/0273-1177(90)90301-F
[14]	Farah A (2008) Comparison of GPS/Galileo single frequency ionospheric models with vertical TEC maps. Artif Satell 43: 75–90. https://doi.org/10.2478/v10018-009-0008-5 doi: 10.2478/v10018-009-0008-5
[15]	Radicella SM, Zhang ML (1995) The improved DGR analytical model of electron density height profile and total electron content in the ionosphere. Ann Geophys 25: 35–41. https://doi.org/10.4401/ag-4130 doi: 10.4401/ag-4130
[16]	Fedrizzi M, de Paula ER, Kantor IJ, et al. (2002) Mapping the low-latitude ionosphere with GPS. GPS World 13: 41–47.
[17]	Krishna KS, Ratnam DV, Sridhar M, et al. (2020) Performance Evaluation of Adjusted Spherical Harmonics Ionospheric Model Over Indian Region. IEEE Access 8: 172610–172622. https://doi.org/10.1109/ACCESS.2020.3024920 doi: 10.1109/ACCESS.2020.3024920
[18]	Sikirica N, Dimc F, Jukic O, et al. (2021) A Risk Assessment of Geomagnetic Conditions Impact on GPS Positioning Accuracy Degradation in Tropical Regions Using Dst Index. In Proceedings of the 2021 International Technical Meeting of The Institute of Navigation, 606–615. https://doi.org/10.33012/2021.17852
[19]	Montenbruck O, González Rodríguez B (2020) NeQuick-G performance assessment for space applications. GPS Solut 24: 1–12. https://doi.org/10.1007/s10291-019-0931-2 doi: 10.1007/s10291-019-0931-2

Reader Comments

Your name:*

Email:*
© 2022 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)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Geosciences

0.8

Metrics

Article views(2168) PDF downloads(96) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Geosciences

Applicability of NeQuick G ionospheric model for single-frequency GNSS users over India

Related Papers:

Abstract

1. Introduction

2. Concepts of Mohand transform

3. Methodology

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

3.2. VITM

4. Problem 1

4.1. The implementation of q-HATM

4.2. The implementation of VITM

5. Graphical and tables discussion

6. Conclusions

Use of Generative-AI tools declaration

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. Concepts of Mohand transform

3. Methodology

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

3.2. VITM

4. Problem 1

4.1. The implementation of q-HATM

4.2. The implementation of VITM

5. Graphical and tables discussion

6. Conclusions

Use of Generative-AI tools declaration

Conflict of interest

References

AIMS Geosciences

Applicability of NeQuick G ionospheric model for single-frequency GNSS users over India

Related Papers:

Abstract

1. Introduction

2. Concepts of Mohand transform

3. Methodology

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

3.2. VITM

4. Problem 1

4.1. The implementation of q-HATM

4.2. The implementation of VITM

5. Graphical and tables discussion

6. Conclusions

Use of Generative-AI tools declaration

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Concepts of Mohand transform

3. Methodology

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

3.2. VITM

4. Problem 1

4.1. The implementation of q-HATM

4.2. The implementation of VITM

5. Graphical and tables discussion

6. Conclusions

Use of Generative-AI tools declaration

Conflict of interest

References