Residual power series scheme treatments for fractional Klein-Gordon problem arising in soliton theory

Article type: Research Article

Keywords: Confluent Bernoulli polynomials, Residual power series scheme, Fractional derivatives, Klien-Gordon equations, Numerical results, Mathematics and computing, Computational science, Pure mathematics

Authors: Saad Z. Rida, Anas A. M. Arafa, Hussein S. Hussein, Ismail Gad Ameen, Marwa M. M. Mostafa

Affiliations: https://ror.org/00jxshx33grid.412707.70000 0004 0621 7833Department of Mathematics, Faculty of Science, South Valley University, Qena, 83523 Egypt; https://ror.org/01wsfe280grid.412602.30000 0000 9421 8094Department of Mathematics, College of Science, Qassim University, Buraydah, Saudi Arabia

License: © The Author(s) 2024 CC BY 4.0 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Article links: DOI: 10.1038/s41598-024-79247-9  |  PubMed: 39715758  |  PMC: PMC11666727

Relevance: Relevant: mentioned in keywords or abstract

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Introduction

Fractional calculus (FC) is a powerful and effective tool for solving many differential equations. It is a branch of mathematical analysis that deals with the generalization of fractional integrals and derivatives. Over the past few decades, FC has attracted much interest due to its applications in fields of sciences. Many phenomena in various science are modeled by fractional partial differential equations (FPDEs) for example mathematical physics, plasma physics, optics, engineering, quantum evolution of complex systems, chemical physics, fractional dynamics, biology, electromagnetic waves, astrophysics, wave propagation phenomenon, and image processing^1–8.

Many researchers have been interested in solving FPDEs by many effective and explicit methods. Researchers have presented many numerical and analytical methods to study these equations such as Adomian decomposition Laplace transform method (ADLTM)^9,10, finite element method¹¹, finite Hankel transform procedure¹², collocation method¹³, Adomian decomposition general transform method¹⁴, Q-homotopy analysis transform method¹⁵, Sawi transform with the homotopy perturbation method¹⁶, modified auxiliary equation method¹⁷, Shehu transform and ADM¹⁸, homotopy perturbation general transform method¹⁹, \(exp(\Phi (\zeta ))\)-expansion method²⁰ and Adomian decomposition formable transform method²¹.

Among these techniques, the RPSS is characterized by simplicity and power as it was first introduced by El-Ajou to find the solution of fuzzy differential equations of the first and second order²². RPSS does not require comparison of the coefficients of the corresponding terms and the recurrence relation. It can be used directly for the target problem by choosing a suitable initial guess approximation. The solution of RPSS is based on obtaining power series for linear and nonlinear problems without discretization, perturbation or linearization^22,23. Over the past few years, RPSS has been applied to solve many ordinary and linear and nonlinear partial differential equations^24–30.

KGP is one of the important mathematical physics models that describe dispersive wave phenomena. In addition, it has many applications such as nonlinear wave propagation, plasma physics, nonlinear optics, fluid dynamics, quantum field theory, relativistic physics, dispersive wave phenomena, condensed matter physics, chemical kinetics and solid-state physics^31–36. The nonlinear fractional time KGP is defined as follows³⁷:

with to initial conditions (ICs) and boundary conditions (BCs):

where \(\Psi ( z ,\tau )\) is wave displacement at \(z\) and \(\tau\), \(\mathfrak {F}_1 ( z )\) and \(\mathfrak {F}_1 ( z )\) are wave kinks, \(\mathbb {H}( z ,\tau )\) is the source function and \(\rho ,\ \sigma ,\ \varrho \ \in \textbf{R}\).

Several numerical and analytical techniques are discussed for solving KGP. For instance, Haar wavelets method³⁸, spline collocation method³⁹, natural homotopy perturbation method^40,41, Cubic trigonometric B-spline basis functions³⁷, Tension spline approach⁴², radial basis functions (RBF)⁴³, clique polynomial method⁴⁴, Bernoulli wavelet collocation method (BWCM)⁴⁵, Sinc Chebyshev collocation method⁴⁶, Clique polynomials⁴⁷, variational iteration method⁴⁸, Sobolev Gradien⁴⁹, \(\bigg (\dfrac{G’}{G}\bigg )\) expansion method⁵⁰ and homotopy analysis transform method⁵¹.

The primary aim of this paper is to present semi analytical technique in order to find the solution for FPDEs in general that is considered new accurate technique and can be applied to obtain the solution for many physical and engineering applications containing FPDEs in a few straightforward steps. Moreover, we utilized CBCA-RPSS for solving KGP which defined in Eqs. (1) and (2). Hence, we compared error norms obtained by CBCA-RPSS with other existing techniques in the literature. As far as we know, CBCA-RPSS has never been used to solve KGP, which is what motivated us to carry out this study.

This current study is designed in the following manner: In “Essential definitions” section, we introduced some mathematical definitions which are utilized in this paper. Main results are presented in “Main results” section. In “Description of methodology” section, we explained the CBCA-RPSS for solving non-linear time-fractional KGP. The proposed problems and norm errors are presented in “The proposed problems and norm errors” section. To illustrate the efficiency and accuracy for CBCA-RPSS, the numerical results and discussion are given in “Results and discussion” section. Finally, “Concluding remarks” section is provided concluding remarks.

Essential definitions

In this section, we present Caputo’s fractional partial derivative (CFPD) and some of its major features. Also essential concepts of fractional power series (FPS) and Bernoulli polynomials are revisited.

Definition 1

The CFPD \(\mathcal {D}^{\zeta }\) of order \(\zeta\) for the given function \(\mathcal {U}( z ,\tau )\) is defined as^5,52:

The CFD satisfies linear property similar to integer order differentiation:

where \(C_1, C_2\) are constants. For CFD, we obtain the major properties as below:

where \(\lceil \zeta \rceil\) denote to the smallest integer greater than or equal to \(\zeta\), where \(\mathcal {N}_{0}= \{0,1,2,…\}\).

Definition 2

The sl-truncated series \(v_{s l}( z ,\tau )\) of the RPSS is given as below^24,53:

and

Definition 3

Bernoulli orthonormal polynomial of order n can be defined in⁵⁴ as:

According to the orthogonal property, we obtain:

Definition 4

The confluent Bernoulli polynomials is described as below⁵⁵:

where \(\mathbb {B}_i\) is Bernoulli numbers are known that \(\mathbb {B}_0=1, \ \mathbb {B}_1=\dfrac{-1}{2},\ \mathbb {B}_2=\dfrac{1}{6};\ \mathbb {B}_{2i+1}=0\ (i=1,2,…)\).

The function \(\Psi ( z ,\tau )\) is defined in terms of confluent Bernoulli polynomials as:

In practice, we truncate the infinite series up to \((n+1)\) terms of confluent Bernoulli polynomials as follows:

Main results

Here, we conclude an approximate formula for CFD of function \(\Psi _n( z ,\tau )\), convergence, uniqueness and error analysis as following theories:

Fractional derivative for confluent Bernoulli polynomials

Theorem 1

Assume that \(\Psi _{n}( z ,\tau )\) be series approximation of confluent Bernoulli which defined by Eq. (11), then \(\mathcal {D}^{\zeta } _{ z } \ \Psi _{n}( z ,\tau )\) is given as:

where \(\mathcal {E}_{i,k} ^{(\zeta )}\) is defined as:

Proof: From the linear properties of CFD, we get

By appling Eqs. (4) and (5), we have \(\mathcal {D}^{\zeta } _{ z }\bigg (\mathfrak {B}^{(a,b)}_i( z )\bigg )=0,\quad i=0,1,2,… \lceil \zeta \rceil -1,\quad \zeta >0\).

Also, \(\forall \ i=\lceil \zeta \rceil , \lceil \zeta \rceil +1,…,n\), we get

by substituting from Eq. (14) in Eq. (13), we obtain

Convergence, uniqueness and error analysis

Theorem 2

Let \(\textbf{H}\) be a Hilbert space. Then the series solution \(\displaystyle \sum _{i=0}^\infty \lambda _{i}(\tau )\ \mathfrak {B}^{(a,b)}_i( z )\) of Eqs. (1) and (2) converges if there exist \(\Delta >0\), such that \(\Arrowvert \lambda _{i+1}(\tau )\ \mathfrak {B}^{(a,b)}_{i+1}( z )\Arrowvert \le \Delta \Arrowvert \lambda _{i}(\tau )\ \mathfrak {B}^{(a,b)}_{i}( z )\Arrowvert\).

Proof

We define sequence of partial sums as: \(\mathbb {V}_{0}= \lambda _{0}(\tau )\ \mathfrak {B}^{(a,b)}_0( z )\),

\(\mathbb {V}_{1}= \lambda _{0}(\tau )\ \mathfrak {B}^{(a,b)}_0( z )+\lambda _{1}(\tau )\ \mathfrak {B}^{(a,b)}_1( z ),\)

\(\mathbb {V}_{2}= \lambda _{0}(\tau )\ \mathfrak {B}^{(a,b)}_0( z )+\lambda _{1}(\tau )\ \mathfrak {B}^{(a,b)}_1( z )+\lambda _{2}(\tau )\ \mathfrak {B}^{(a,b)}_2( z ),\)

. . .

\(\mathbb {V}_{i}= \lambda _{0}(\tau )\ \mathfrak {B}^{(a,b)}_0({z})+\lambda _{1}(\tau )\ \mathfrak {B}^{(a,b)}_1( z )+\lambda _{2}(\tau )\ \mathfrak {B}^{(a,b)}_2({z})+…+\lambda _{i}(\tau )\ \mathfrak {B}^{(a,b)}_i({z}).\)

We need to prove that \(\{\mathbb {V}_{i}\}_{i=0}^{\infty }\) is convergent Cauchy sequence. Moreover, we consider

\(\Arrowvert \mathbb {V}_{i+1}- \mathbb {V}_{i}\Arrowvert =\Arrowvert \lambda _{i+1}(\tau )\ \mathfrak {B}^{(a,b)}_{i+1}({z})\Arrowvert \le \Delta \Arrowvert \lambda _{i}(\tau )\ \mathfrak {B}^{(a,b)}_{i}({z})\Arrowvert \le \Delta ^2\Arrowvert \lambda _{i-1}(\tau )\ \mathfrak {B}^{(a,b)}_{i-1}({z})\Arrowvert …\le \Delta ^{i+1}\Arrowvert \lambda _{0}(\tau )\ \mathfrak {B}^{(a,b)}_{0}({z})\Arrowvert\).

For \(i,\mu \in \mathcal {N},i> \mu\), it yields

By using triangle inequality, we get

\(\Arrowvert \mathbb {V}_{i}- \mathbb {V}_{i-1}+\mathbb {V}_{i-1}-\mathbb {V}_{i-2}+…+\mathbb {V}_{\mu +1}-\mathbb {V}_{\mu }\Arrowvert \le \Arrowvert \mathbb {V}_{i}- \mathbb {V}_{i-1}\Arrowvert +\Arrowvert \mathbb {V}_{i-1}- \mathbb {V}_{i-2}\Arrowvert +…+\Arrowvert \mathbb {V}_{\mu +1}- \mathbb {V}_{\mu }\Arrowvert .\)

Further, we obtain

Hence, \(\{\mathbb {V}_{i}\}_{i=0}^{\infty }\) is convergent cauchy sequence in Hilbert space. \(\square\)

The uniqueness theory of solutions is one of important theorems in the qualitative analysis of models and has been mentioned in many references (see e.g.^24,56). Therefore, we state it as follow:

Theorem 3

Let \(\{\Psi _n\}\) is a sequence of solution such that \(\Psi _n\) is converging at \(n\rightarrow \infty\) then solution is unique.

Proof

Assume \(\bar{\Psi }\) an approximate solution of the proposed problem and \(\Psi _n\) is a general solution, then let \(\Psi _{n}( z ,\tau )\) has two limits of convergence as \(\Psi _n\rightarrow \bar{\Psi }\) and \(\Psi _n\rightarrow \mathbb {U}\) as \(n\rightarrow \infty\), then suppose that \(\bar{\Psi }\ne \mathbb {U}\) and \(\Arrowvert \bar{\Psi }-\mathbb {U}\Arrowvert = \varphi\). Then

since

\(\Psi _n\rightarrow \bar{\Psi }\), and \(\Psi _n\rightarrow \mathbb {U}\), then \(\Arrowvert \Psi _n\rightarrow \bar{\Psi }\Arrowvert \rightarrow 0\) and \(\Arrowvert \Psi _n\rightarrow \mathbb {U}\Arrowvert\) as \(n\rightarrow \infty\).

From Eq. (16), we obtain \(\Arrowvert \bar{\Psi }-\mathbb {U}\Arrowvert \rightarrow 0\) as \(n\rightarrow \infty\).

Hence the solution is unique. \(\square\)

Theorem 4

Assume that \(\mathcal {D}^{i\zeta } \ \Psi (\tau ) \in \mathcal {C}[0,1],\ \forall \ i=0,1,…,n+1\ and\ 0<\zeta \le 1\). Let \(\Psi _{n}(\tau )\) is the best square approximation of \(\Psi (\tau )\), then :

where \(\mathscr {A}=\max _{\tau \in [0,1]}\ \mathcal {D}^{(n+1)\zeta }\Psi (\tau )\) and \(\mathscr {B}= \max \{\tau _0,\tau -\tau _0\}\).

Proof

By using Taylor’s expansion for the function \(\Psi (\tau )\) as:

Suppose

we have \(|{\Psi (\tau )- \tilde{\Psi }_{n}(\tau )}|=|{\dfrac{(\tau -\tau _0)^{(n+1) \zeta }}{\Gamma (1+(n+1) \zeta )}\mathcal {D}^{(n+1)\zeta } \Psi (\epsilon )}|.\)

Since, \(\Psi _{n}(\tau )\) is the best square approximation for \(\Psi (\tau )\), we get

Let \(\mathscr {B}= \max \{\tau _0,\tau -\tau _0\}\), then we get

Then, by taking the square roots of both sides, we obtain

\(\square\)

Description of methodology

In this section, we utilize confluent Bernoulli collocation approach for obtaining approximate solution of non-linear time-fractional KGP which defined in Eqs. (1) and (2), as following steps:

\(\blacklozenge\)\(\mathbf {Step \ I.}\) By substituting Eqs. (11) and (15) into Eq. (1), we have

\(\blacklozenge\) \(\mathbf {Step \ II.}\) Now, we collocate Eq. (18) at \(z _{r},\) \(r = 0,1,2,…n-\lceil \zeta \rceil\) and the collocation point of confluent Bernoulli \(\mathfrak {B}^{(a,b)}_{n+1-\lceil \zeta \rceil } ( z )\), we get system of fractional order differential equations (FODEs) as:

\(\blacklozenge\) \(\mathbf {Step \ III.}\) By appling Eq. (11) into Eq. (2) at \(z _{r}\), we obtain system of algebraic equations as:

where BCs

To obtain the unknown coefficients \(\lambda _i(\tau )\), \(i=0,1,2,…,n\) combing Eqs. (19), (20) and (21), we have system of FODEs, which can be solved by utilizing RPSS.

To determine the unknown coefficients of \(\lambda _i(\tau )\), we take \(n=2\), \(a=2,b=1\) in Eq. (19):

By solving Eq. (21), then we get

By substituting Eq. (23) into Eq. (22), then

RPSS assumes the solution of Eq. (25) using FPS at \(\tau _0=0\) as:

Next, let \(\lambda _{0(\kappa ,l)}(\tau )\) indicate the \(\kappa ^{th}\) truncated series of \(\lambda _{0}(\tau )\) which take the form:

where \(\Omega _{0}\) and \(\Omega _{1}\) can be obtained by solving Eqs. (20) and (21). The RPSS assumes the solution of Eq. (25) using FPS at \(\tau _0=0\) as:

and

The proposed problems and norm errors

To investigate the accuracy and applicability of CBCA-RPSS, we presented numerical outcomes through an error norms \(\mathscr {L}_{\infty }\), \(\mathscr {L}_{2}\) and root mean square error (RMSE) which defined in⁵⁷ as follows:

where \(\texttt{E}_\iota =|{ \Psi _{\iota }(Exact\ solution)-\Psi _{\iota }(Approximate \ solution)}|\).

Problem 1. Consider non-linear time-fractional KGP^44,45

with to ICs and BCs:

which has an exact solution at \(\zeta =2\) as \(\Psi ( z ,\tau )= z cos(\tau ).\)

According to an explanation of the proposed method in “Description of methodology” section, then the approximate solution is:

\(\Psi _{2}( z ,\tau ) = \lambda _{0}(\tau )+ (2 z -\dfrac{1}{2})\lambda _{1}(\tau ) + (3 z ^2-2 z +\dfrac{1}{6})\lambda _{2}(\tau ),\)

where \(\lambda _{0}(\tau )=\dfrac{1}{4}-\dfrac{1}{4}\dfrac{\tau ^{\zeta }}{\Gamma (1+\zeta )}+\dfrac{1}{4}\dfrac{\tau ^{2\zeta }}{\Gamma (1+2\zeta )}-…,\)

and

Problem 2. Consider non-linear time-fractional KGP⁴⁵

with to ICs and BCs:

which has an exact solution at \(\zeta =2\) as \(\Psi ( z ,\tau )= z sin(\dfrac{\pi \tau }{2}).\)

By appling the proposed method, then we can obtain the approximate solution as: \(\Psi _{2}( z ,\tau ) = \lambda _{0}(\tau )+ (2 z -\dfrac{1}{2})\lambda _{1}(\tau ) + (3 z ^2-2 z +\dfrac{1}{6})\lambda _{2}(\tau ),\)

where \(\lambda _{0}(\tau )=\dfrac{\pi }{8}\tau -\dfrac{\pi ^{3}}{32}\dfrac{\tau ^{\zeta +1}}{\Gamma (2+\zeta )}+\dfrac{\pi ^{5}}{128}\dfrac{\tau ^{2\zeta +1}}{\Gamma (2+2\zeta )}-…\),

and

Problem 3. Consider the non-linear equation⁴⁵

with to ICs and BCs:

which has an exact solution at \(\zeta =1\) as \(\Psi ( z ,\tau )= z sin(\tau ^{2}).\) According to the proposed method, then we can write the approximate solution as:

\(\Psi _{2}( z ,\tau ) = \lambda _{0}(\tau )+ (2 z -\dfrac{1}{2})\lambda _{1}(\tau ) + (3 z ^2-2 z +\dfrac{1}{6})\lambda _{2}(\tau ),\)

where \(\lambda _{0}(\tau )=\dfrac{1}{2}\dfrac{\tau ^{2\zeta }}{\Gamma (1+2\zeta )}-30 \dfrac{\tau ^{6\zeta }}{\Gamma (1+6\zeta )}+7560\dfrac{\tau ^{10\zeta }}{\Gamma (1+10\zeta )}-…\),

and

Results and discussion

In this section, we present an obtained numerical outcomes for all problems through tables and graphics in two and three dimensions.

In Table 1, we compare the \(\mathscr {L}_{\infty }\), \(\mathscr {L}_{2}\) and RMS error norms obtained by CBCA-RPSS at \(\zeta =2\) with Tension spline approach \(O(k^2+k^2h^2+h^2)\) method⁴², \(O(k^2+k^2h^2+h^4)\) method⁴², RBF approximation⁴³, Clique polynomial method⁴⁴ and BWCM⁴⁵ of Problem 1. Figure 1 displays exact solution and CBCA-RPSS solution with its absolute error of Problem 1. Figure 2 expresses the comparison of CBCA-RPSS solutions for various fractional-order of \(\zeta\) while Fig. 3 demonstrate the behavior of fractional-order \(\zeta\) on CBCA-RPSS solution with exact solution at \(\tau =2\) in 2D graph of Problem 1.

Table 2 display a comparison of error norms for the approximate solution by CBCA-RPSS at \(\zeta =2\) with that Tension spline approach \(O(k^2+k^2h^2+h^2)\) method⁴², \(O(k^2+k^2h^2+h^4)\) method⁴², RBF approximation⁴³ and BWCM⁴⁵ of Problem 2. Figure 4 displays exact solution and CBCA-RPSS solution with its absolute error of Problem 2. Figure 5 expresses the comparison for CBCA-RPSS solution of various fractional-order of \(\zeta\) while Fig. 6 show the behavior of fractional-order \(\zeta\) on CBCA-RPSS solution with exact solution at \(\tau =0.1\) in 2D graph of Problem 2.

In Table 3, the error norms are reported for \(\zeta =1\) and compared with BWCM⁴⁵ while in Table 4 show comparison of absolute error of CBCA-RPSS at \(\zeta =1\) and \(z =0.1\) with that BWCM⁴⁵, ADM with the Bernstein polynomials (BPs)⁵⁸ and ADM with modified Bernstein polynomials (MBPs)⁵⁸ of Problem 3. Figure 7 displays exact solution and CBCA-RPSS solution with its absolute error of Problem 3. Figure 8 expresses the comparison for CBCA-RPSS solution of various fractional-order of \(\zeta\) while Fig. 9 demonstrate the behavior of fractional-order \(\zeta\) on CBCA-RPSS solution with exact solution at \(\tau =1\) in 2D graph of Problem 3.

Concluding remarks

In this paper, CBCA-RPSS was successfully used to obtain an approximate solution for non-linear time-fractional KGP. First, by utilizing confluent Bernoulli polynomial and their properties the non-linear time-fractional KGP is reduced into a system of FODEs. Secondly, the RPSS is used to solve these system of equations with helping of the given initial and boundary conditions. We derived some theories of convergence, uniqueness and error analysis of proposed method. Finally, the comparisons of results abtained by CBCA-RPSS were performed and found that it is very computationally accurate than other existing techniques in literature^42–45,58. Also, this comparisons shows that present method is very straightforward, simple, accurate and can be applied to solve nonlinear PDE problems that arise in engineering problems and complex phenomena.

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