Bounds of random star discrepancy for HSFC-based sampling

Xiaoda Xu; Xiaoda Xu

doi:10.3934/math.2025255

AIMS Mathematics

2025, Volume 10, Issue 3: 5532-5551. doi: 10.3934/math.2025255

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Research article

Bounds of random star discrepancy for HSFC-based sampling

Xiaoda Xu ^,

School of Mathematics and Science, Suqian University, Suqian 223800, China

Received: 12 October 2024 Revised: 09 February 2025 Accepted: 24 February 2025 Published: 12 March 2025
MSC : 11K38, 65C10, 65D30

This paper is dedicated to the estimation of the probabilistic upper bounds of star discrepancy for Hilbert's space filling curve (HSFC) sampling. The primary concept revolves around the stratified random sampling method, with the relaxation of the stringent requirement for a sampling number $N = m^d$ in jittered sampling. We leverage the benefits of this sampling method to achieve superior results compared to Monte Carlo (MC) sampling. We also provide applications of the main result, which pertain to weighted star discrepancy, $L_2$ -discrepancy, integration approximation in certain function spaces and examples in finance.

Keywords:

Citation: Xiaoda Xu. Bounds of random star discrepancy for HSFC-based sampling[J]. AIMS Mathematics, 2025, 10(3): 5532-5551. doi: 10.3934/math.2025255

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Abstract

1. Introduction

In the modelling of numerous real-world events in control theory, chemistry, physics and other branches of engineering and science, fractional derivatives can be used successfully ^[1,2,3,4,5]. The basic reason for this is that fractional calculus can be used to model real-world issues successfully because realistic models not only depend on the present time but also on the past historical time ^[6,7,8]. As a result, the fractional differential equations (FDEs) have drawn the interest of numerous scientific and engineering academics due to their significant role in understanding a number of real-life process that happen in the natural sciences, such as mechanical systems, wave propagation phenomena, earthquake modelling, image processing and control theory. Fractional calculus can be used to describe and reformulate these processes as FDEs. The use of FDEs in the aforementioned and other phenomena is notable for their nonlocality feature. Similarly, the differential operators give an excellent way to describe the memory and inherited characteristics of distinct processes and materials ^[9,10].

In the framework of fractional derivatives, partial differential equations (PDEs) are regarded as a potential tool in mathematical modelling to explain and understand some physical processes structures that are complicated due to outside influences. Because of this reason, researchers utilized them to both construct a natural problem that is easily accessible and to simplify the regulating plan without losing some inherited material or memorial impact ^[1,2,3]. The development of trustworthy numerical methods to handle the fraction PDEs of physical concern has also been the subject of countless efforts, many of which have been successful ^{[14,15,16,17,18,19]}. Many real-world issues, such as earthquakes, gas dynamics, traffic flow and oscillation, can be formulated as nonlinear PDEs in the framework of fractional derivative, and the solutions of fractional PDEs provide remarkable insight into the behaviour of particular dynamic systems ^[4,5]. Therefore, it is important to develop a practical and useful method for identifying analytical solutions to these and other issues. To examine and develop analytical-approximate solutions of fractional FDEs and PDEs, researchers have recently used a variety of analytical and numerical methodologies, such as the Homotopy perturbation transform method ^[22], residual power series (RPS) method ^[23], Adomian decomposition method ^[24], variational iteration transform method ^[25,26], Iterative laplace transform method ^[27], Elzaki transform decomposition method ^[28,29] and Natural transform decomposition method ^[30,31].

A numerical analytic method for solving many forms of ordinary, partial, integro-differential equations and fractional fuzzy differential equations is known as the residual power series (RPS) technique. The RPSM was introduced by Jordanian mathematician Omar Abu Arqub in 2013 ^[32]. Due to the fact that it offers closed-form solutions of well-known functions, it is an efficient optimization strategy ^[33,34]. With the use of the fractional residual power series (FRPS) technique, fuzzy FDEs, a variety of FDEs, and integral equations having fractional-order are solved. Such as, Coupled fractional resonant Schrödinger equations ^[35], time-fractional Fokker-Planck models ^[36], fractional fredholm integro-differential equations ^[37], fractional Newell-hitehead-Segel equation ^[38], singular initial value problems ^[39], Fractional partial differential equations ^[40], fractional Kundu-Eckhaus and massive Thirring equations ^[41] and several classes of fractional fuzzy differential equations ^[42,43]. The Yang transform (YT) is a powerful tool for resolving numerous complex models that are appearing in various branches of the natural sciences. When analytical methods are combined with the YT operator, non-linear problems can be solved more quickly and with more precision.

Researchers introduced a new approach to solve fractional differential equations by combining two well-known approaches. Several of these groups include a combination of the homotopy analysis method and the natural transform ^[44], also homotopy perturbation approach and the Sumudu transform ^[45], the Yang transform and the Adomian decomposition method ^[46], and the Laplace transform with RPSM ^[47]. The main goal of this work is to examine the approximate solution of nonlinear systems by implementing the Yang residual power series (YRPS) method. The YT and RPS approaches are combined in the YRPS approach, which provides both approximate and accurate solutions as quickly fractional power series (FPS) solutions. The proposed system is converted to Yang space, and then the solutions in the form of algebraic equations are created. Lastly, the Yang inverse is applied to the proposed problem results. In contrast to the FRPS approach, which depends on the fractional derivative and consume time to compute the various derivatives of fractional-order in steps of determining the solutions, the unknown coefficients in a modified Yang expansion can be identified by employing the limit idea. The YRPS approach take less time and provide higher accuracy with minor computational requirements.

In this article, we consider fractional nonlinear systems as:

$\begin{equation} \begin{split} &D_{t}^{\varsigma}{u}+{v}_{{\vartheta}}{w}_{{\psi}}-{v}_{{\psi}}{w}_{{\vartheta}}+{u} = 0, \ \ \ 0 < \varsigma\leq1, \\ &D_{t}^{\varsigma}{v}+{w}_{{\vartheta}}{u}_{{\psi}}-{u}_{{\psi}}{w}_{{\vartheta}}-{v} = 0, \\ &D_{t}^{\varsigma}{w}+{u}_{{\vartheta}}{v}_{{\psi}}-{u}_{{\psi}}{v}_{{\vartheta}}-{w} = 0, \\ \end{split} \end{equation}$

(1.1)

having initial sources

$\begin{equation} \begin{split} &{u}({\vartheta}, {\psi}, 0) = f_{0}({\vartheta}, {\psi}), \\ &{v}({\vartheta}, {\psi}, 0) = g_{0}({\vartheta}, {\psi}), \\ &{w}({\vartheta}, {\psi}, 0) = h_{0}({\vartheta}, {\psi}), \\ \end{split} \end{equation}$

(1.2)

and

$\begin{equation} \begin{split} &D_{t}^{\varsigma}{u}-{u}_{{\vartheta}{\vartheta}}-2{u}{u}_{{\vartheta}}+({u}{v})_{{\vartheta}} = 0, \ \ \ 0 < \varsigma\leq1, \\ &D_{t}^{\varsigma}{v}-{v}_{{\vartheta}{\vartheta}}-2{v}{v}_{{\vartheta}}+({u}{v})_{{\vartheta}} = 0, \\ \end{split} \end{equation}$

(1.3)

having initial sources

$\begin{equation} \begin{split} &{u}({\vartheta}, {\psi}, 0) = f_{0}({\vartheta}, {\psi}), \\ &{v}({\vartheta}, {\psi}, 0) = g_{0}({\vartheta}, {\psi}).\\ \end{split} \end{equation}$

(1.4)

Following is the breakdown of our study. Section 2 reviews the YT, as well as some fundamental definitions and theorem pertaining to fractional calculus. Section 3 describes the idea behind the suggested technique for creating the approximation of the fractional model taken into account in Eq (1.1). In Section 4, the YRPS methodology is applied to solve fractional nonlinear systems in order to show the applicability and efficacy of the method in analysing the solutions of time-PDEs of fractional order. Section 5 concludes with a summary of our results.

2. Preliminaries

In this part, we review some definitions and fractional derivatives theorems in Caputo manner along with YT properties.

Definition 2.1. In Caputo sense the fractional derivative of a function ${u}({\vartheta}, {t})$ is given as ^[48]

$\begin{equation} ^{C}D_{{t}}^{{\varsigma}}{u}({\vartheta}, {t}) = J_{{t}}^{m-{\varsigma}}{u}^{{m}}({\vartheta}, {t}), \ \ m-1 < \varsigma\leq m, \ \ t > 0, \end{equation}$

(2.1)

with $m\in N$ and $J_{{t}}^{\varsigma}$ represents the fractional integral in Riemann-Liouville manner of ${u}({\vartheta}, {t})$ as

$\begin{equation} J_{t}^{\varsigma} u(\vartheta, t) = \frac{1}{\Gamma(\varsigma)} \int_{0}^{t}(t-\tau)^{\varsigma-1} u(\vartheta, \tau) d \tau. \end{equation}$

(2.2)

Definition 2.2. The YT of a function $\phi(\kappa)$ is stated by $\mathcal{Y}\{u(t)\}$ or M(s) as ^[49,50]

$\begin{equation} \mathcal{Y}\{u(t)\} = M(s) = \int_0^{\infty}e^{\frac{-t}{s}}u(t)dt, \ \ t > 0, \ \ s\in(-t_1, t_2). \end{equation}$

(2.3)

The inverse YT is given as

$\begin{equation} \mathcal{Y}^{-1}\{M(s)\} = u(t). \end{equation}$

(2.4)

Definition 2.3. The nth derivatives YT is given ^[49,50]

$\begin{equation} \mathcal{Y}\{u^{n}(t)\} = \frac{M(s)}{s^{n}}-\sum\limits_{k = 0}^{n-1}\frac{u^{k}(0)}{s^{n-k-1}}, \ \ \forall \ \ n = 1, 2, 3, \cdots \end{equation}$

(2.5)

Definition 2.4. The YT of derivatives having order fraction is as ^[49,50]

$\begin{equation} \mathcal{Y}\{u^{\varsigma}(t)\} = \frac{M(s)}{s^{\varsigma}}-\sum\limits_{k = 0}^{n-1}\frac{u^{k}(0)}{s^{\varsigma-(k+1)}}, \ \ 0 < \varsigma \leq n. \end{equation}$

(2.6)

Theorem 2.5. The fractional power series $\sum_{m-0}^{\infty} a_m(t-\varepsilon)^{m p}$ can converge within just three of the following ways:

(1) The series converges only when the radius of convergence equals zero, or $t = \varepsilon$ .

(2) The series converges with a radius of convergence equal to $\infty$ for all $t \geq \varepsilon$ .

(3) The series diverges for $t > c+R$ and converges for $\varepsilon \leq t < \varepsilon+R$ and some real positive integer $R$ .

In this context, $R$ refers to the fractional power series radius of convergence.

3. Idea of YRPS

In this part, we will present the general implementation of YRPS for solving fractional nonlinear systems of PDEs.

By employing YT to Eq (1.1), we have

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s)-{s}f_{0}({\vartheta}, {\psi})+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{U}]\right] = 0, \\ &V({\vartheta}, {\psi}, s)-{s}g_{0}({\vartheta}, {\psi})+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{V}]\right] = 0, \\ &W({\vartheta}, {\psi}, s)-{s}h_{0}({\vartheta}, {\psi})+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{W}]\right] = 0.\\ \end{split} \end{equation}$

(3.1)

Considering that the solution of Eq (3.1) has the appropriate expansion

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s) = \sum\limits_{n = 0}^{\infty}s^{n\varsigma+1}f_{n}({\vartheta}, {\psi}, s), \ \ V({\vartheta}, {\psi}, s) = \sum\limits_{n = 0}^{\infty}s^{n\varsigma+1}g_{n}({\vartheta}, {\psi}, s), \\ &W({\vartheta}, {\psi}, s) = \sum\limits_{n = 0}^{\infty}s^{n\varsigma+1}h_{n}({\vartheta}, {\psi}, s).\\ \end{split} \end{equation}$

(3.2)

The $k^{th}$ -truncated series are

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s) = {s}f_{0}({\vartheta}, {\psi}, s)+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}f_{n}({\vartheta}, {\psi}, s), \ \ V({\vartheta}, {\psi}, s) = {s}g_{0}({\vartheta}, {\psi}, s)+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}g_{n}({\vartheta}, {\psi}, s), \\ &W({\vartheta}, {\psi}, s) = {s}h_{0}({\vartheta}, {\psi}, s)+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}h_{n}({\vartheta}, {\psi}, s).\ \ \ k = 1, 2, 3, 4\cdots \end{split} \end{equation}$

(3.3)

By Yang residual functions (YRFs)

$\begin{equation} \begin{split} \mathcal{Y}_{{t}}Res_{u}({\vartheta}, {\psi}, s) = &U({\vartheta}, {\psi}, s)-{s}f_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{U}]\right], \\ \mathcal{Y}_{{t}}Res_{v}({\vartheta}, {\psi}, s) = &V({\vartheta}, {\psi}, s)-{s}g_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{V}]\right], \\ \mathcal{Y}_{{t}}Res_{\rho}({\vartheta}, {\psi}, s) = &W({\vartheta}, {\psi}, s)-{s}h_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{W}]\right].\\ \end{split} \end{equation}$

(3.4)

And the ${k}^{th}$ -YRFs as:

$\begin{equation} \begin{split} \mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, {\psi}, s) = &U_{k}({\vartheta}, {\psi}, s)-{s}f_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]+\mathcal{Y}_{t}^{-1}[{U}_{k}]\right], \\ \mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, {\psi}, s) = &V_{k}({\vartheta}, {\psi}, s)-{s}g_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]-\mathcal{Y}_{t}^{-1}[{V}_{k}]\right], \\ \mathcal{Y}_{{t}}Res_{w, k}({\vartheta}, {\psi}, s) = &W_{k}({\vartheta}, {\psi}, s)-{s}h_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]-\mathcal{Y}_{t}^{-1}[{W}_{k}]\right].\\ \end{split} \end{equation}$

(3.5)

To highlight some facts, the YRPSM contains the following characteristics:

● $\mathcal{Y}_{{t}}Res({\vartheta}, {\psi}, s) = 0$ and $\lim_{{j}\rightarrow \infty}\mathcal{Y}_{{t}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = \mathcal{Y}_{{{t}}}Res_{{u}}({\vartheta}, {\psi}, s)$ for each $s > 0.$

● $\lim_{s\rightarrow \infty}s\mathcal{Y}_{{{t}}}Res_{{u}}({\vartheta}, {\psi}, s) = 0\Rightarrow \lim_{s\rightarrow \infty}s\mathcal{Y}_{{{t}}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = 0.$

● $\lim_{s\rightarrow \infty}s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = \lim_{s\rightarrow \infty}s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = 0, \ \ 0 < {\varsigma}\leq 1, \ \ {k} = 1, 2, 3, \cdots$ .

Now to determine the coefficients $f_{n}({\vartheta}, {\psi}, s)$ , $g_{n}({\vartheta}, {\psi}, s)$ , $h_{n}({\vartheta}, {\psi}, s)$ and $l_{n}({\vartheta}, {\psi}, s)$ , we resolve the below system recursively as

$\begin{equation} \begin{split} &\lim\limits_{s\rightarrow \infty}s^{{k}{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = 0, \ \ {k} = 1, 2, \cdots, \\ &\lim\limits_{s\rightarrow \infty}s^{{k}{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{v}, {k}}({\vartheta}, {\psi}, s) = 0, \ \ {k} = 1, 2, \cdots, \\ &\lim\limits_{s\rightarrow \infty}s^{{k}{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{\rho}, {k}}({\vartheta}, {\psi}, s) = 0, \ \ {k} = 1, 2, \cdots.\\ \end{split} \end{equation}$

(3.6)

Finally by employing inverse YT to Eq (3.3), to obtain the ${k}^{th}$ analytical solutions of ${u}_{{k}}({\vartheta}, {\psi}, {t})$ , ${v}_{{k}}({\vartheta}, {\psi}, {t})$ and ${\rho}_{{k}}({\vartheta}, {\psi}, {t})$

4. Applications

In this part, we find the solution of fractional nonlinear systems by implementing the suggested approach.

4.1. Problem

Let us assume the system of fractional PDEs:

$\begin{equation} \begin{split} &D_{t}^{\varsigma}{u}+{v}_{{\vartheta}}{w}_{{\psi}}-{v}_{{\psi}}{w}_{{\vartheta}}+{u} = 0, \\ &D_{t}^{\varsigma}{v}+{w}_{{\vartheta}}{u}_{{\psi}}-{u}_{{\psi}}{w}_{{\vartheta}}-{v} = 0, \\ &D_{t}^{\varsigma}{w}+{u}_{{\vartheta}}{v}_{{\psi}}-{u}_{{\psi}}{v}_{{\vartheta}}-{w} = 0.\\ \end{split} \end{equation}$

(4.1)

By considering Eq (4.1), having below initial sources:

$\begin{equation} \begin{split} &{u}({\vartheta}, {\psi}, 0) = e^{{\vartheta}+{\psi}}, \\ &{v}({\vartheta}, {\psi}, 0) = e^{{\vartheta}-{\psi}}, \\ &{w}({\vartheta}, {\psi}, 0) = e^{{\psi}-{\vartheta}}.\\ \end{split} \end{equation}$

(4.2)

By employing YT to Eq (4.1) and using Eq (4.2), we have

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s)-{s}e^{{\vartheta}+{\psi}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{U}]\right] = 0, \\ &V({\vartheta}, {\psi}, s)-{s}e^{{\vartheta}-{\psi}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{V}]\right] = 0, \\ &W({\vartheta}, {\psi}, s)-{s}e^{{\psi}-{\vartheta}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{W}]\right] = 0.\\ \end{split} \end{equation}$

(4.3)

The $k^{th}$ -truncated term series are

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s) = {s}e^{{\vartheta}+{\psi}}+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}f_{n}({\vartheta}, {\psi}, s), \ \ V({\vartheta}, {\psi}, s) = {s}e^{{\vartheta}-{\psi}}+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}g_{n}({\vartheta}, {\psi}, s), \\ &W({\vartheta}, {\psi}, s) = {s}e^{{\psi}-{\vartheta}}+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}h_{n}({\vartheta}, {\psi}, s), \ \ \ k = 1, 2, 3, 4\cdots \end{split} \end{equation}$

(4.4)

and the ${k}^{th}$ -YRFs as:

$\begin{equation} \begin{split} \mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, {\psi}, s) = &U_{k}({\vartheta}, {\psi}, s)-{s}e^{{\vartheta}+{\psi}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]+\mathcal{Y}_{t}^{-1}[{U}_{k}]\right], \\ \mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, {\psi}, s) = &V_{k}({\vartheta}, {\psi}, s)-{s}e^{{\vartheta}-{\psi}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]-\mathcal{Y}_{t}^{-1}[{V}_{k}]\right], \\ \mathcal{Y}_{{t}}Res_{w, k}({\vartheta}, {\psi}, s) = &W_{k}({\vartheta}, {\psi}, s)-{s}e^{{\psi}-{\vartheta}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]-\mathcal{Y}_{t}^{-1}[{W}_{k}]\right].\\ \end{split} \end{equation}$

(4.5)

To find $f_{k}({\vartheta}, {\psi}, s)$ , $g_{k}({\vartheta}, {\psi}, s)$ and $h_{k}({\vartheta}, {\psi}, s)$ $k = 1, 2, 3, \cdots,$ the $kth$ -truncated series equation Eq (4.4) will be inserted into the $kth$ -Yang residual function equation Eq (4.5), which will then be multiplied by $s^{k{\varsigma}+1}$ to solve the relation recursively $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, {\psi}, s)) = 0$ , $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, {\psi}, s)) = 0$ , and $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{w, k}({\vartheta}, {\psi}, s)) = 0$ , $k = 1, 2, 3, \cdots$ .

Few terms are as:

$\begin{equation} \begin{split} f_{1}({\vartheta}, {\psi}, s) = &e^{{\vartheta}+{\psi}}, \ \ g_{1}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{1}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{2}({\vartheta}, {\psi}, s) = &-e^{{\vartheta}+{\psi}}, \ \ g_{2}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{2}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{3}({\vartheta}, {\psi}, s) = &e^{{\vartheta}+{\psi}}, \ \ g_{3}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{3}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{4}({\vartheta}, {\psi}, s) = &-e^{{\vartheta}+{\psi}}, \ \ g_{4}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{4}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{5}({\vartheta}, {\psi}, s) = &e^{{\vartheta}+{\psi}}, \ \ g_{5}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{5}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{6}({\vartheta}, {\psi}, s) = &-e^{{\vartheta}+{\psi}}, \ \ g_{6}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{6}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \end{split} \end{equation}$

(4.6)

and go on.

By substituting the values of $f_{k}({\vartheta}, {\psi}, s)$ , $g_{k}({\vartheta}, {\psi}, s)$ and $h_{k}({\vartheta}, {\psi}, s)$ , $k = 1, 2, 3, \cdots,$ in Eq (4.4), we have

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s) = {s}e^{{\vartheta}+{\psi}}-{s^{\varsigma+1}}e^{{\vartheta}+{\psi}}+{s^{2\varsigma+1}}e^{{\vartheta}+{\psi}}-{s^{3\varsigma+1}}e^{{\vartheta}+{\psi}}+{s^{4\varsigma+1}}e^{{\vartheta}+{\psi}}-{s^{5\varsigma+1}}e^{{\vartheta}+{\psi}}+{s^{6\varsigma+1}}e^{{\vartheta}+{\psi}}+\cdots, \\ &V({\vartheta}, {\psi}, s) = {s}e^{{\vartheta}-{\psi}}+{s^{\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{2\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{3\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{4\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{5\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{6\varsigma+1}}e^{{\vartheta}-{\psi}}+\cdots, \\ &W({\vartheta}, {\psi}, s) = {s}e^{{\psi}-{\vartheta}}+{s^{\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{2\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{3\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{4\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{5\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{6\varsigma+1}}e^{{\psi}-{\vartheta}}+\cdots.\\ \end{split} \end{equation}$

(4.7)

By employing inverse YT, we have

$\begin{equation*} \begin{split} &u({\vartheta}, {\psi}, {t}) = e^{{\vartheta}+{\psi}}\left(1-\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}-\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}-\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right), \\ &v({\vartheta}, {\psi}, {t}) = e^{{\vartheta}-{\psi}}\left(1+\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}+\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}+\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right), \\ &\rho({\vartheta}, {\psi}, {t}) = e^{{\psi}-{\vartheta}}\left(1+\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}+\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}+\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right).\\ \end{split} \end{equation*}$

Putting $\varsigma = 1$

$\begin{equation} \begin{split} &u({\vartheta}, {\psi}, {t}) = e^{{\vartheta}+{\psi}}\left(1-{t}+\frac{{t}^{2}}{2!}-\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}-\frac{{t}^{5}}{5!}+\cdots\right), \\ &v({\vartheta}, {\psi}, {t}) = e^{{\vartheta}-{\psi}}\left(1+{t}+\frac{{t}^{2}}{2!}+\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}+\frac{{t}^{5}}{5!}+\cdots\right), \\ &w({\vartheta}, {\psi}, {t}) = e^{{\psi}-{\vartheta}}\left(1+{t}+\frac{{t}^{2}}{2!}+\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}+\frac{{t}^{5}}{5!}+\cdots\right).\\ \end{split} \end{equation}$

(4.8)

Thus we get exact solutions as

$\begin{equation} \begin{split} &u({\vartheta}, {\psi}, {t}) = e^{{\vartheta}+{\psi}-{t}}, \\ &v({\vartheta}, {\psi}, {t}) = e^{{\vartheta}-{\psi}+{t}}, \\ &\rho({\vartheta}, {\psi}, {t}) = e^{{\psi}-{\vartheta}+{t}}.\\ \end{split} \end{equation}$

(4.9)

4.2. Problem

Let us assume the system of fractional Burger's equations:

$\begin{equation} \begin{split} &D_{t}^{\varsigma}{u}-{u}_{{\vartheta}{\vartheta}}-2{u}{u}_{{\vartheta}}+({u}{v})_{{\vartheta}} = 0, \\ &D_{t}^{\varsigma}{v}-{v}_{{\vartheta}{\vartheta}}-2{v}{v}_{{\vartheta}}+({u}{v})_{{\vartheta}} = 0.\\ \end{split} \end{equation}$

(4.10)

By considering Eq (4.10), having below initial sources:

$\begin{equation} \begin{split} &{u}({\vartheta}, 0) = \sin({{\vartheta}}), \\ &{v}({\vartheta}, 0) = \sin({{\vartheta}}).\\ \end{split} \end{equation}$

(4.11)

By employing YT to Eq (4.10) and using Eq (4.11), we get

$\begin{equation} \begin{split} &U({\vartheta}, s)-{s}\sin({{\vartheta}})+{s^{\varsigma}}\mathcal{Y}_{t}\left[-\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}{\vartheta}}]-2\mathcal{Y}_{t}^{-1}[{U}]\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{(UV)}_{{\vartheta}}]\right] = 0, \\ &V({\vartheta}, s)-{s}\sin({{\vartheta}})+{s^{\varsigma}}\mathcal{Y}_{t}\left[-\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}{\vartheta}}]-2\mathcal{Y}_{t}^{-1}[{V}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{(UV)}_{{\vartheta}}]\right] = 0.\\ \end{split} \end{equation}$

(4.12)

The $k^{th}$ -truncated series are

$\begin{equation} \begin{split} &U({\vartheta}, s) = {s}\sin({{\vartheta}})+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}f_{n}({\vartheta}, s), V({\vartheta}, s) = {s}\sin({{\vartheta}})+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}g_{n}({\vartheta}, s), k = 1, 2, 3, 4\cdots \end{split} \end{equation}$

(4.13)

and the ${k}^{th}$ -YRFs as:

$\begin{equation} \begin{split} \mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, s) = &U_{k}({\vartheta}, s)-{s}\sin({{\vartheta}})+{s^{\varsigma}}\mathcal{Y}_{t}\left[-\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}{\vartheta}, {k}}]-2\mathcal{Y}_{t}^{-1}[{U}_{k}]\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}, {k}}]+\mathcal{Y}_{t}^{-1}[{(UV)}_{{\vartheta}, {k}}]\right], \\ \mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, s) = &V_{k}({\vartheta}, s)-{s}\sin({{\vartheta}})+{s^{\varsigma}}\mathcal{Y}_{t}\left[-\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}{\vartheta}, {k}}]-2\mathcal{Y}_{t}^{-1}[{V}_{k}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]+\mathcal{Y}_{t}^{-1}[{(UV)}_{{\vartheta}, {k}}]\right].\\ \end{split} \end{equation}$

(4.14)

To find $f_{k}({\vartheta}, s)$ and $g_{k}({\vartheta}, s)$ , $k = 1, 2, 3, \cdots,$ the $kth$ -truncated series equation Eq (4.13) will be inserted into the $kth$ -Yang residual function equation Eq (4.14), which will then be multiplied by $s^{k{\varsigma}+1}$ to solve the relation recursively $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, s)) = 0$ and $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, s)) = 0$ , $k = 1, 2, 3, \cdots$ .

Few terms are as:

$\begin{equation} \begin{split} f_{1}({\vartheta}, s) = &\sin({{\vartheta}}), \ \ g_{1}({\vartheta}, s) = \sin({{\vartheta}}), \\ f_{2}({\vartheta}, s) = &-\sin({{\vartheta}}), \ \ g_{2}({\vartheta}, s) = -\sin({{\vartheta}}), \\ f_{3}({\vartheta}, s) = &\sin({{\vartheta}}), \ \ g_{3}({\vartheta}, s) = \sin({{\vartheta}}), \\ f_{4}({\vartheta}, s) = &-\sin({{\vartheta}}), \ \ g_{4}({\vartheta}, s) = -\sin({{\vartheta}}), \\ f_{5}({\vartheta}, s) = &\sin({{\vartheta}}), \ \ g_{5}({\vartheta}, s) = \sin({{\vartheta}}), \\ f_{6}({\vartheta}, s) = &-\sin({{\vartheta}}), \ \ g_{6}({\vartheta}, s) = -\sin({{\vartheta}}), \end{split} \end{equation}$

(4.15)

and so on.

By substituting the values of $f_{k}({\vartheta}, s)$ and $g_{k}({\vartheta}, s)$ , $k = 1, 2, 3, \cdots,$ in Eq (4.13), we have

$\begin{equation} \begin{split} &U({\vartheta}, s) = {s}\sin({{\vartheta}})-{s^{\varsigma+1}}\sin({{\vartheta}})+{s^{2\varsigma+1}}\sin({{\vartheta}})-{s^{3\varsigma+1}}\sin({{\vartheta}})+{s^{4\varsigma+1}}\sin({{\vartheta}})-{s^{5\varsigma+1}}\sin({{\vartheta}})+{s^{6\varsigma+1}}\sin({{\vartheta}})+\cdots, \\ &V({\vartheta}, s) = {s}\sin({{\vartheta}})-{s^{\varsigma+1}}\sin({{\vartheta}})+{s^{2\varsigma+1}}\sin({{\vartheta}})-{s^{3\varsigma+1}}\sin({{\vartheta}})+{s^{4\varsigma+1}}\sin({{\vartheta}})-{s^{5\varsigma+1}}\sin({{\vartheta}})+{s^{6\varsigma+1}}\sin({{\vartheta}})+\cdots.\\ \end{split} \end{equation}$

(4.16)

By employing inverse YT, we have

$\begin{equation*} \begin{split} &u({\vartheta}, {t}) = \sin({{\vartheta}})\left(1-\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}-\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}-\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right), \\ &v({\vartheta}, {t}) = \sin({{\vartheta}})\left(1-\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}-\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}-\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right).\\ \end{split} \end{equation*}$

Putting $\varsigma = 1$

$\begin{equation} \begin{split} &u({\vartheta}, {t}) = \sin({{\vartheta}})\left(1-{t}+\frac{{t}^{2}}{2!}-\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}-\frac{{t}^{5}}{5!}+\cdots\right), \\ &v({\vartheta}, {t}) = \sin({{\vartheta}})\left(1-{t}+\frac{{t}^{2}}{2!}-\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}-\frac{{t}^{5}}{5!}+\cdots\right).\\ \end{split} \end{equation}$

(4.17)

Thus we get accurate solutions as

$\begin{equation} \begin{split} &u({\vartheta}, {t}) = e^{-{t}}\sin({{\vartheta}}), \\ &v({\vartheta}, {t}) = e^{-{t}}\sin({{\vartheta}}).\\ \end{split} \end{equation}$

(4.18)

4.3. Results and discussion

In the present study, we obtained the closed form solution of two nonlinear fractional systems by means of YRPSM upto sixth order. The obtained results are describes with the help of graphs and table. The YRPSM results for system $1$ on the basis of error is illustrated in Table 1. Figure 1a, b shows the exact and analytical behavior whereas , shows the 3-D and 2-D behavior of approximate solution at different fractional-orders for ${u}({\vartheta}, {\psi}, {t})$ . Figure 2a, b shows the exact and analytical behavior whereas , shows the 3-D and 2-D behavior of approximate solution at different fractional-orders for ${v}({\vartheta}, {\psi}, {t})$ . Figure 3a, b shows the exact and analytical behavior whereas , shows the 3-D and 2-D behavior of approximate solution at different fractional-orders for ${w}({\vartheta}, {\psi}, {t})$ of system 1. Similarly, Figure 4a, b shows the exact and analytical behavior whereas , shows the 3-D and 2-D behavior of approximate solution at different fractional-orders for ${u}({\vartheta}, {t})$ and ${v}({\vartheta}, {t})$ of system 2. The behavior of Table and Figures shows that our solution is in good agreement with the exact solutions of the problems.

Table 1. Behavior on the basis of Error by implementing YRPSM for the suggested problem at different values of t and

${\vartheta}$ with

${\varsigma = 1}$ ,

${\psi} = 1$ .

${t}$	${\vartheta}$	$\|{u}_{exact}-{u}_{YRPSM}\|$	$\|{v}_{exact}-{v}_{YRPSM}\|$	$\|{w}_{exact}-{w}_{YRPSM}\|$
0.2	0.2	2.671 $\times10^{-8}$	9.927 $\times10^{-8}$	1.8080 $\times10^{-8}$
	0.4	3.263 $\times10^{-8}$	1.212 $\times10^{-8}$	1.4810 $\times10^{-8}$
	0.6	3.986 $\times10^{-7}$	1.481 $\times10^{-8}$	1.2126 $\times10^{-7}$
	0.8	4.868 $\times10^{-7}$	1.808 $\times10^{-7}$	9.9270 $\times10^{-7}$
	1	5.945 $\times10^{-6}$	2.209 $\times10^{-7}$	8.1280 $\times10^{-6}$
0.4	0.2	2.1267 $\times10^{-7}$	7.981 $\times10^{-8}$	1.4544 $\times10^{-8}$
	0.4	2.5975 $\times10^{-7}$	9.748 $\times10^{-7}$	1.1908 $\times10^{-7}$
	0.6	3.1727 $\times10^{-6}$	1.190 $\times10^{-7}$	9.7489 $\times10^{-7}$
	0.8	3.8752 $\times10^{-6}$	1.454 $\times10^{-6}$	7.9817 $\times10^{-6}$
	1	4.7330 $\times10^{-5}$	1.776 $\times10^{-6}$	6.5349 $\times10^{-6}$
0.6	0.2	7.142 $\times10^{-6}$	2.707 $\times10^{-7}$	4.9333 $\times10^{-6}$
	0.4	8.7233 $\times10^{-6}$	3.3068 $\times10^{-6}$	4.0390 $\times10^{-6}$
	0.6	1.0654 $\times10^{-5}$	4.0390 $\times10^{-5}$	3.3068 $\times10^{-5}$
	0.8	1.3013 $\times10^{-5}$	4.9333 $\times10^{-5}$	2.7074 $\times10^{-5}$
	1	1.5894 $\times10^{-4}$	6.0254 $\times10^{-5}$	2.2166 $\times10^{-5}$

| Show Table

DownLoad: CSV

Figure 1. 3-D and 2-D behavior for

${u}({\vartheta}, {\psi}, t)$ of Problem

$1$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. 3-D and 2-D behavior for

${v}({\vartheta}, {\psi}, t)$ of Problem

$1$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. 3-D and 2-D behavior for

${w}({\vartheta}, {\psi}, t)$ of Problem

$1$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. 3-D and 2-D behavior for

${u}({\vartheta}, t)$ and

${v}({\vartheta}, t)$ of Problem

$2$ .

DownLoad: Full-Size Img PowerPoint

5. Conclusions

To find the solutions of DEs and FDEs, a variety of numerical and analytical techniques are employed, some of which are regarded by the prospect of non-exact solutions. This work emphasised that the suggested strategy, YRPS, is an easy-to-use analytical technique for developing accurate and approximative solutions for certain systems of FDEs with suitable initial sources. The aforementioned method gave us the solutions in the Yang transform space by making it simple to calculate the expansion series constants with the aid of the limit concept at infinity. In contrast to the RPS approach, the YRPS method requires minimal calculations to obtain the series coefficients since it uses the limit idea rather than the fractional derivative. The calculated results demonstrate how closely the approximative solutions approach the accurate solution. This proves that the aforementioned method is an appropriate and extraordinarily effective method for obtaining the approximative and analytical solutions to a large number of linear and non-linear fractional problems that arise in engineering and applied physics.

Funding

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that they have no competing interests.

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Bounds of random star discrepancy for HSFC-based sampling

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2. Preliminaries

3. Idea of YRPS

4. Applications

4.1. Problem

4.2. Problem

4.3. Results and discussion

5. Conclusions

Funding

Acknowledgments

Conflict of interest

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AIMS Mathematics

Bounds of random star discrepancy for HSFC-based sampling

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Abstract

1. Introduction

2. Preliminaries

3. Idea of YRPS

4. Applications

4.1. Problem

4.2. Problem

4.3. Results and discussion

5. Conclusions

Funding

Acknowledgments

Conflict of interest

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