Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions

Bin Jebreen, Haifa

doi:https://doi.org/10.1155/2023/8453459

Advances in Mathematical Physics

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 8453459 | https://doi.org/10.1155/2023/8453459

Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions

Haifa Bin Jebreen¹

Academic Editor: S. A. Edalatpanah

Received23 Aug 2023

Revised12 Nov 2023

Accepted25 Nov 2023

Published12 Dec 2023

Abstract

The focus of this paper is on utilizing the spectral element method to find the numerical solution of the fractional Klein–Gordon equation. The algorithm employs interpolating scaling functions (ISFs) that meet specific properties and satisfy the multiresolution analysis. Using an orthonormal projection, the equation is mapped to the scaling spaces in this method. A matrix representation of the Caputo fractional derivative of ISFs is presented using matrices representing the fractional integral and derivative operators. Using this matrix, the spectral element method reduces the desired equation to a system of algebraic equations. To find the solution, the generalized minimal residual method (GMRES method) and Newton’s method are used in linear and nonlinear forms of this system, respectively. The method’s convergence is proven, and some illustrative examples confirm it. The method is characterized by its simplicity in implementation, high efficiency, and significant accuracy.

1. Introduction

Our aim in this study is to implement and develop the Galerkin method to approximate the solution of the fractional Klein–Gordon equation:subjected to the following conditions:where and are constants, is a continuous function, and is a known function that fulfills the following Lipschitz condition:where is referred to the Lipschitz constant. The fractional derivative is of the Caputo fractional derivative (CFD) type, such that we shall introduce it in the sequel.

For many years, positive integer derivatives have been used in partial differential equations (PDEs). But, in recent years, we have noticed a trend toward using fractional derivatives to model physical phenomena. It seems like there are many advantages to this approach, including greater accuracy and more flexibility in the types of systems that can be modeled. Meanwhile, some applications of these equations can be mentioned such as colored noise [1], bioengineering [2–4], solid mechanics [5], continuum and statistical mechanics [6], earthquakes [7], anomalous transport [8], economics [9], fluid-dynamic traffic model [10], and engineering and natural sciences [11–13]. There are some analytical methods to solve these types of equations [14–16]. However, when the equations become more complicated, these methods no longer work. So, numerical approaches can address this deficiency. Here, we mention some of these methods, including finite difference method [17], collocation method [18, 19], Galerkin method [20, 21], finite element method [22], kernel-based pseudo-spectral method [23], nonuniform difference schemes [24], and fractional differential transform method [25].

It is common knowledge that the Klein–Gordon equations, both linear and nonlinear, are widely used when it comes to modeling various physical phenomena. They have been used to model everything from solitons to condensed matter physics, as well as classical and quantum mechanics. In 1926, two physicists named Klein [26] and Gordon [27] introduced this equation to describe relativistic electrons. According to the importance of these equations, studying and finding their solution can be very useful. To this end, through the utilization of different numerical techniques, we have able to solve these equations, such as the He’s method [28, 29], radial basis functions [30], decomposition method [31], meshless method [32], Galerkin method [33], Tau method [34], differential transform method [35], b-spline scaling function [36], and Adomians decomposition scheme [37].

The fractional Klein–Gordon equation has been garnering significant attention from scientists as of late. In fact, fractional Klein–Gordon equation is the generalization of fractional Klein–Gordon equation of integer order. On the other hand, this equation describes nonlocal relationships in space and time using power-law memory kernels. According to the definition of fractional derivative, it is worth noting that the fractional derivative at a point depends on all values of the function. Thus, it is expected that the fractional derivative operation involves some sort of boundary conditions, involving information on the function further out [38]. As we know, when the fractional order approaches an integer, the fractional derivative converges to the ordinary derivative. So, we can conclude that the fractional differential equations are close to the real modeling of phenomena and the differential equations of integer order are the limits of them. Several numerical schemes have been developed by mathematicians to aid in solving this equation. It is remarkable to witness the rapid advancements being made in this field. The Sinc–Chebyshev based collocation method described in Nagy’s [39] study is an effective means of solving the nonlinear version of this equation. I came across a fascinating article by Golmankhaneh et al. [40] on using the homotopy perturbation method to solve the problem. In Singh et al.’s [41] study, the authors employed Chebyshev polynomials of the third kind to implement the collocation method for finding the solution to this problem. Cubic B-spline functions are used to solve this problem by a numerical method [42]. One of the approach that has been found to be efficient for finding numerical solutions for equations of this type is the meshless method suggested by Gharian et al. [43]. The problem at hand may be resolved through the utilization of generalized polynomials, as presented in Hassani et al.’s [44] study.

Interpolating scaling functions (ISFs) are a family of powerful bases that emerge from a multiresolution analysis (MRA). These bases are orthonormal and by dilation and translation generate a family of nested subspaces of . These bases are also essential for constructing Alpert’s multiwavelets. These bases were first introduced by Alpert et al. [45] where PDEs were solved by an adaptive method using these bases. ISFs have shown their adaptability and usefulness in a wide range of mathematical problems, such as solving ordinary differential equations [46, 47], PDEs [48], integral equations [49], and more.

In this paper, we use the spectral element method based on ISFs to solve the fractional Klein–Gordon equation. The fundamental idea behind this method is to consider the unknown solution as a linear combination of basis functions. Then, using the operational matrix of CFD and spectral element method, the desired equation reduces to a system of algebraic equations. Considering the flexibility of the ISFs in the selection of and parameters, the presented method solves this type of equation with appropriate accuracy. As you know, the main challenges in equations such as the Klein–Gordon equation are the nonlinearity and existence of the fractional derivative. When we use other types of bases, these two factors cause some limitations in implementing the spectral element method. However, since there is no need to calculate the integral in using the ISFs to approximate the functions, the problems of nonlinearity can be easily solved. About the fractional derivatives, we know that the equation may have a nonsmooth solution near the boundaries, which many numerical methods fail to overcome. The scaling functions, due to the properties that they inherit from multiresolution analysis, by increasing the scale parameter s can overcome this challenge.

In this article, we have organized the information into different sections. Section 2 provides an introduction to ISFs and their properties, as well as a matrix representation of CFD. In Section 3, we explain and execute the proposed method, which utilizes the spectral element method. Additionally, we present a theorem that demonstrates the convergence of this method. Section 4 contains results from various numerical experiments. Finally, we complete this work through the inclusion of a conclusion in Section 5.

2. Interpolating Scaling Bases

Before implementing the algorithm of the proposed scheme, it is worthwhile to give the bases’ construction briefly. To this end, according to our knowledge of these bases, these bases are piecewise polynomials of the degree less than a constant number, called multiplicity parameter [45, 50, 51]. The main configuration of these bases is made by Lagrange polynomials , which are used in their construction from the roots of Legendre polynomials . Suppose that these roots are indicated by . Furthermore, assume that the Gauss–Legendre quadrature weights , , determined by , is given. The ISFs are specified by the following equation:

Motivated by -inner product, the ISFs consist of orthonormal bases for the finite-dimensional space:

By dividing the interval into subintervals , (namely ) in which is given and , we are now ready to introduce the subspaces that generate a MRA. Recall that MRA consists of nested subspaces that satisfy some conditions cf [52]. Here, we designate these subspaces by the following equation:where and denote 9the translation and dilation operators, respectively, and are determined by the following equation:

To approximate any function , using an orthonormal projection , it can be mapped onto via a finite sum:where is a fixed integer number and the coefficients can be computed by the following equation:

As you see, computing these integrals lead to high computational cost, so to avoid this, the interpolating property of the bases could be a problem solver. This property gives rise to calculating these coefficients without integration, i.e.:where . Let us determine by a vector function , consisting of all scaling functions in . According to this presentation, Equation (9) may be rewritten as follows:where the elements of -dimensional vector are determined by , in which . The superscript “T” in this text indicates the meaning of “transpose.”

There is an error in this approximation. In order to determine an estimate for the maximum error, we can refer to the Lemma 1, [50].

Lemma 1. Given , assume that be of class , times continuously differentiable function. The error of approximation (9) is estimated by the following equation:

To extend the approximation to the 2D space , which is generated by , the projection can be utilized to approximate the 2D function , viz.:in which the coefficients and , are computed by the following equation:

But, similar to the 1D case, to avoid calculating these integrals, they can be found using the following equation:

Lemma 2 (cf Theorem 1, Saray et al. [53]). Given , assume that be a sufficiently smooth function. The approximation error (14) can be limited by the following equation:where:

2.1. Matrix Representation of CFD

Before introducing a matrix representation of CFD (), it is necessary to determine some preliminaries about fractional calculus. Additionally, since the matrix representation of the CFD is derived from the matrix representation of the fractional integral (FI), it is necessary to incorporate this matrix.

Definition 1. Given , the FI operator of order is specified by the following equation:where indicates the Gamma function.

Given the power function, its fractional integration is also a power function, that is:

The norm of the FI operator is bounded, as motivated by Kilbas et al. [54].

Lemma 3. There is an estimation of the bound of the fractional integral operator in , viz.:

The matrix representation of the fractional integral operator is denoted by . In order to determine , it is necessary to once again estimate the fractional integral of the vector function using ISFs, viz.:where is a square matrix. The objective is to identify the elements present in this matrix. To proceed, we have to take help from the following lemma.

Lemma 4 (cf, Asadzadeh and Saray [55]). The Lagrange polynomials for nodes may be written as follows:where and

Based on Equation (11), it is possible to compute the elements of matrix using the following equation:where and , , . The elements of matrix is previously obtained by Asadzadeh and Saray [55]. They proved this matrix is an upper triangular matrix in which entries can be computed by considering the three following cases.(1)Given , we can easily prove that:(2)Considering , setting , by the change of variable , Equation (26) leads to the following equation:According to the closed form ISFs and Lemma 4, we can obtaine the following equation:Now, the elements situated on the diagonal are specified by the following equation:where states the following hypergeometric function:Furthermore, the beta function is denoted by , and .(3)Given , we have . So, it can be show that:Using the same argument as in Case (2), we get the following equation:Applying the binomial expansion of as follows:the entries of above the main diagonal can be specified the following equation:in which , .

Assume that is a space of functions such that:

As we know, if , then the CFD:exists for almost every .

Now, our objective is to find an operational matrix for the operator such that it satisfies the following equation:

To find this matrix, motivated by Equation (37), the operational matrix can be utilized as follows:where the matrix D is used to represent the derivative operation, as explained in Alpert et al.’s [45] study. Consequently, the matrix can be determined by the following equation:

3. Methods Description

Recall that our objective is to introduce and implement an effective numerical algorithm for Equation (1). The approach used in this numerical algorithm involves implementing the Galerkin method. The method begins by expanding the unknown solution based on ISFs, viz.:or equivalently:in which we have a square matrix, , of order , with coefficients that are currently unknown.

Substituting Equation (42) into Equation (1) leads to the following equation:

Using Equation (42) and applying the operational matrices and , we can approximate all terms of Equation (43), as follows.(i)Taking CFD from Equation (41), and using the matrix , one can write the following equation:(ii)For the second one, namely, , we use the matrix , as follows:(iii)Given , let us imagine a matrix that has dimensions and meets the following condition:Note that the entries of the matrix consist of the linear or nonlinear equations of the unknowns .(iv)Using the operator , the given function can also map into the space , viz.:in which is an -dimensional square matrix.

Putting Equations (44)–(47) back into Equation (43) gives rise to introducing the residual:where . Applying the Galerkin method requires fulfilling the following equation:or equivalent to the following equation:

Because the ISFs are orthonormal bases, the aforementioned equation leads to a linear or nonlinear system:

Now, the functions , , , and must be mapped to space to apply the boundary and initial conditions. So, we have the following equation:

By rewriting Equation (51) using the following equation:we have a new system:

To gain the unknown matrix , the generalized minimal residual method (GMRES method) [56] and Newton’s method are used to determine the unknown vector for the linear and nonlinear forms, respectively.

3.1. Convergence Verification

Theorem 1. Assume that the functions and are sufficiently smooth. Furthermore, if satisfies the Lipschitz condition (4), thus the presented method for Equation (1) is convergent when the parameters or tend to infinity.

Proof. Subtracting Equation (1) from the following equation:the residual is determined by the following equation:Taking the norm from both sides and using the triangle inequality, we get the following equation:Given . By utilizing Lemmas 2, 3 and Equation (37), it can be verified that all expressions (57) have bounds, viz.:(i)For expression , we have the following equation:where is a constant and can be obtained by replacing with in Lemma 2.(ii)Employing Lemma 2 and the Lipschitz condition (4) gives rise to the following equation:where is constant.(iii)For other expressions, using Lemma 2, it is straightforward to establish the following equation:To proceed, by substituting Equations (58)–(60) in Equation (57), we obtain the following equation:in which . Therefore:

4. Illustrative Numerical Examples

This section includes some examples to showcase the effectiveness of the proposed method. To illustrate the results and make a global view of the present method and its efficiency, sometimes, the absolute errors:and error:are reported in tables or plotted in figures.

Example 1. We dedicate the first example to the following equation:subjected to the following conditions:in which: provides the precise solution for this equation [57].
The approximate solution and corresponding absolute error for different choices of , taking and , are plotted in Figure 1. Figure 2 illustrates the accuracy of the presented method with , , and . The -error is also tabulated in Table 1 with different choices of , taking and . From our observation, the results clearly showcase the effectiveness and precision of the method that was presented. To compare the present method with the method presented in Dehghan et al.’s [57] study and show the efficiency and accuracy of the method, Table 2 is reported.

(a)

(b)

(c)

(d)

Example 2. We devote this example to the following equation:with the following boundary and initial conditions:The precise solution for this equation is assigned by [58].
To provide evidence of the method’s accuracy, Table 3 is tabulated. According to our analysis, when the parameter increases, the error must be reduced. Table 4 is reported to show the accuracy at different points. To this end, we can find the absolute error in this table. Compared to other existing methods, it is worth mentioning that the proposed scheme provides better accuracy than the high-order difference method [58]. This superiority is reported in Table 5 with different eta. The approximate solution and absolute error are illustrated in Figure 3 for different .

(a)

(b)

(c)

(d)

Example 3. The third example is dedicated to the nonlinear fractional Klein–Gordon equation:with the following boundary and initial conditions:The precise solution for this equation is assigned by [59].
By setting , , and , the -errors and maximum absolute error are reported in Table 6 at different times. To compare the presented method with others, Table 7 is reported. Compared to other methods including clique polynomials [59], tension spline approach [60] and radial basis functions [30], the presented method gives better accuracy. Figure 4 is provided to demonstrate the approximate solution and corresponding absolute error using the presented method with , , and .

5. Conclusion

Due to the use of fractional derivatives and nonlinearity, solving the fractional Klein–Gordon equation is a challenging task. This work uses the spectral element method to solve the problem by mapping it to an approximation subspace of through an orthonormal projection. The ISFs that possess specific properties and satisfy the MRA are made use in this method. To this end, a matrix representation of the CFD of ISFs is presented. We prove the method is convergent, and several examples have been prepared to verify the findings of this investigation. The results speak for themselves, and it is evident that this method presents the effective scheme that solve the problem in the accuracy manner. Additionally, when compared to other methods, it outperforms them in terms of accuracy and speed. These findings are truly remarkable and show the immense potential of this approach. The method proposed here has the potential to solve both fractional and nonfractional equations with ease. Its simplicity in implementation, combined with its high efficiency and significant accuracy, make it a strong candidate for solving the same equations.

To summarize, the following key points can be mentioned:(i)The method has a simple and practical structure, making it easy to extend for solving various fractional equations.(ii)The presented method can solve any kind of equation due to its flexibility in choosing scale and multiplicity parameters.(iii)The speed of calculations increases because there is no need for integration due to the properties of the bases used.(iv)Increasing the value of the scale parameter can lead to more appropriate accuracy for problems with nonsmooth solutions at the beginning and end points of the solution interval.

In the future, we plan to extend our numerical approaches for solving generalized fractional models, including the time-fractional diffusion equation [61], the time-fractional mobile–immobile advection–dispersion equation [62], the time-fractional mobile/immobile transport model [63], and multiterm boundary value problem of variable order [64], etc.

Data Availability

The data used to support this study are included in this article.

Conflicts of Interest

The author declares that there is no conflicts of interest regarding the publication of this article

Acknowledgments

This project was supported by Researchers Supporting Project number (RSP2023R210), King Saud University, Riyadh, Saudi Arabia.

References

B. Mandelbrot, “Some noises with spectrum, a bridge between direct current and white noise,” IEEE Transactions on Information Theory, vol. 13, no. 2, pp. 289–298, 1967.
View at: Publisher Site | Google Scholar
R. L. Magin, “Fractional calculus in bioengineering, part 1,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1, pp. 1–104, 2004.
View at: Publisher Site | Google Scholar
R. L. Magin, “Fractional calculus in bioengineering—part 2,” Critical Reviews in Biomedical Engineering, vol. 32, no. 2, pp. 105–193, 2004.
View at: Publisher Site | Google Scholar
R. L. Magin, “Fractional calculus in bioengineering—part 3,” Critical Reviews in Biomedical Engineering, vol. 32, no. 3/4, pp. 194–377, 2004.
View at: Publisher Site | Google Scholar
Y. A. Rossikhin and M. V. Shitikova, “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Applied Mechanics Reviews, vol. 50, no. 1, pp. 15–67, 1997.
View at: Publisher Site | Google Scholar
F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 291–348, Springer, Verlag, New York, 1997.
View at: Google Scholar
J. He, “Nonlinear oscillation with fractional derivative and its applications,” in International Conference on Vibrating Engineering, pp. 288–291, Scientific Research, Dalian, China, 1998.
View at: Google Scholar
R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A: Mathematical and General, vol. 37, no. 31, pp. R161–R208, 2004.
View at: Publisher Site | Google Scholar
R. T. Baillie, “Long memory processes and fractional integration in econometrics,” Journal of Econometrics, vol. 73, no. 1, pp. 5–59, 1996.
View at: Publisher Site | Google Scholar
J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science, Technology & Society, vol. 15, pp. 86–90, 1999.
View at: Google Scholar
H. Günerhan, H. Dutta, M. A. Dokuyucu, and W. Adel, “Analysis of a fractional HIV model with caputo and constant proportional caputo operators,” Chaos, Solitons & Fractals, vol. 139, Article ID 110053, 2020.
View at: Publisher Site | Google Scholar
H. Günerhan, M. K. A. Kaabar, and E. Çelik, “Novel analytical and approximate-analytical methods for solving the nonlinear fractional smoking mathematical model,” Sigma Journal of Engineering and Natural Sciences, vol. 41, no. 2, pp. 331–343, 2023.
View at: Google Scholar
H. Günerhan, H. Rezazadeh, W. Adel et al., “Analytical approximate solution of fractional order smoking epidemic model,” Advances in Mechanical Engineering, vol. 14, no. 9, pp. 1–11, 2022.
View at: Publisher Site | Google Scholar
M. Alquran, H. M. Jaradat, and M. I. Syam, “Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method,” Nonlinear Dynamics, vol. 90, no. 4, pp. 2525–2529, 2017.
View at: Publisher Site | Google Scholar
A. El-Ajou, O. A. Arqub, Z. Al Hour, and S. Momani, “New results on fractional power series: theories and applications,” Entropy, vol. 15, no. 12, pp. 5305–5323, 2013.
View at: Publisher Site | Google Scholar
A. Qazza, R. Saadeh, and E. Salah, “Solving fractional partial differential equations via a new scheme,” AIMS Mathematics, vol. 8, no. 3, pp. 5318–5337, 2022.
View at: Publisher Site | Google Scholar
Y. Zhang, “A finite difference method for fractional partial differential equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 524–529, 2009.
View at: Publisher Site | Google Scholar
S. Bonyadi, Y. Mahmoudi, M. Lakestani, and M. J. Rad, “Numerical solution of space–time fractional PDEs with variable coefficients using shifted Jacobi collocation method,” Computational Methods for Differential Equations, vol. 11, no. 1, pp. 81–94, 2023.
View at: Google Scholar
M. Javidi, “Chebyshev spectral collocation method for computing numerical solution of telegraph equation,” Computational Methods for Differential Equations, vol. 1, no. 1, pp. 16–29, 2013.
View at: Google Scholar
C. Li, Z. Li, and Z. Wang, “Mathematical analysis and the local discontinuous Galerkin method for Caputo–Hadamard fractional partial differential equation,” Journal of Scientific Computing, vol. 85, no. 2, Article ID 41, 2020.
View at: Publisher Site | Google Scholar
Z. Mao and J. Shen, “Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients,” Journal of Computational Physics, vol. 307, pp. 243–261, 2016.
View at: Publisher Site | Google Scholar
N. J. Ford, J. Xiao, and Y. Yan, “A finite element method for time fractional partial differential equations,” Fractional Calculus and Applied Analysis, vol. 14, no. 3, pp. 454–474, 2011.
View at: Publisher Site | Google Scholar
M. Fardi, “A kernel-based pseudo-spectral method for multi-term and distributed order time-fractional diffusion equations,” Numerical Methods for Partial Differential Equations, vol. 39, no. 3, pp. 2630–2651, 2023.
View at: Publisher Site | Google Scholar
M. Fardi, M. A. Zaky, and A. S. Hendy, “Nonuniform difference schemes for multi-term and distributed-order fractional parabolic equations with fractional laplacian,” Mathematics and Computers in Simulation, vol. 206, pp. 614–635, 2023.
View at: Publisher Site | Google Scholar
H. Günerhan and E. Çelik, “Analytical and approximate solutions of fractional partial differential-algebraic equations,” Applied Mathematics and Nonlinear Sciences, vol. 5, no. 1, pp. 109–120, 2020.
View at: Publisher Site | Google Scholar
O. Klein, “Quantentheorie und fünfdimensionale relativitätstheorie,” Zeitschrift für Physik, vol. 37, pp. 895–906, 1926.
View at: Publisher Site | Google Scholar
W. Gordon, “Der comptoneffekt nach der schrödingerschen theorie,” Zeitschrift für Physik, vol. 40, pp. 117–133, 1926.
View at: Publisher Site | Google Scholar
S. Abbasbandy, “Numerical solution of non-linear Klein–Gordon equations by variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 70, no. 7, pp. 876–881, 2007.
View at: Publisher Site | Google Scholar
F. Shakeri and M. Dehghan, “Numerical solution of the Klein–Gordon equation via He’s variational iteration method,” Nonlinear Dynamics, vol. 51, pp. 89–97, 2008.
View at: Publisher Site | Google Scholar
M. Dehghan and A. Shokri, “Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 400–410, 2009.
View at: Publisher Site | Google Scholar
D. Kaya and S. M. El-Sayed, “A numerical solution of the Klein–Gordon equation and convergence of the decomposition method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 341–353, 2004.
View at: Publisher Site | Google Scholar
A. Hussain, S. Haq, and M. Uddin, “Numerical solution of Klein–Gordon and sine-Gordon equations by meshless method of lines,” Engineering Analysis with Boundary Elements, vol. 37, no. 11, pp. 1351–1366, 2013.
View at: Publisher Site | Google Scholar
Y. V. E. S. Tourigny, “Product approximation for nonlinear Klein–Gordon equations,” IMA Journal of Numerical Analysis, vol. 10, no. 3, pp. 449–462, 1990.
View at: Publisher Site | Google Scholar
B. N. Saray, M. Lakestani, and C. Cattani, “Evaluation of mixed Crank–Nicolson scheme and Tau method for the solution of Klein–Gordon equation,” Applied Mathematics and Computation, vol. 331, pp. 169–181, 2018.
View at: Publisher Site | Google Scholar
A. S. V. R. Kanth and K. Aruna, “Differential transform method for solving the linear and nonlinear Klein–Gordon equation,” Computer Physics Communications, vol. 180, no. 5, pp. 708–711, 2009.
View at: Publisher Site | Google Scholar
M. Lakestani and M. Dehghan, “Collocation and finite difference-collocation methods for the solution of nonlinear Klein–Gordon equation,” Computer Physics Communications, vol. 181, no. 8, pp. 1392–1401, 2010.
View at: Publisher Site | Google Scholar
E. Y. Deeba and S. A. Khuri, “A decomposition method for solving the nonlinear Klein–Gordon equation,” Journal of Computational Physics, vol. 124, pp. 442–448, 1996.
View at: Publisher Site | Google Scholar
F. Liu, M. M. Meerschaert, S. Momani, N. N. Leonenko, W. Chen, and O. P. Agrawal, “Fractional differential equations,” International Journal of Differential Equations, vol. 2010, Article ID 215856, 2 pages, 2010.
View at: Publisher Site | Google Scholar
A. M. Nagy, “Numerical solution of time fractional nonlinear Klein–Gordon equation using Sinc–Chebyshev collocation method,” Applied Mathematics and Computation, vol. 310, pp. 139–148, 2017.
View at: Publisher Site | Google Scholar
A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, “On nonlinear fractional Klein–Gordon equation,” Signal Processing, vol. 91, no. 3, pp. 446–451, 2011.
View at: Publisher Site | Google Scholar
H. Singh, D. Kumar, and R. K. Pandey, “An efficient computational method for the time-space fractional Klein–Gordon equation,” Frontiers in Physics, vol. 8, Article ID 281, 2020.
View at: Publisher Site | Google Scholar
M. Amin, M. Abbas, M. K. Iqbal, and D. Baleanu, “Numerical treatment of time-fractional Klein–Gordon equation using redefined extended cubic B-spline functions,” Frontiers in Physics, vol. 8, Article ID 288, 2020.
View at: Publisher Site | Google Scholar
D. Gharian, F. M. M. Ghaini, M. H. Heydari, and Z. Avazzadeh, “A meshless solution for the variable-order time fractional nonlinear Klein–Gordon equation,” International Journal of Applied and Computational Mathematics, vol. 6, no. 5, pp. 1–16, 2020.
View at: Publisher Site | Google Scholar
H. Hassani, J. A. T. Machado, and E. Naraghirad, “An efficient numerical technique for variable order time fractional nonlinear Klein–Gordon equation,” Applied Numerical Mathematics, vol. 154, pp. 260–272, 2020.
View at: Publisher Site | Google Scholar
B. Alpert, G. Beylkin, D. Gines, and L. Vozovoi, “Adaptive solution of partial differential equations in multiwavelet bases,” Journal of Computational Physics, vol. 182, no. 1, pp. 149–190, 2002.
View at: Publisher Site | Google Scholar
S. H. Seyedi, B. N. Saray, and M. R. H. Nobari, “Using interpolation scaling functions based on Galerkin method for solving non-Newtonian fluid flow between two vertical flat plates,” Applied Mathematics and Computation, vol. 269, pp. 488–496, 2015.
View at: Publisher Site | Google Scholar
S. H. Seyedi, B. N. Saray, and A. Ramazani, “High-accuracy multiscale simulation of three-dimensional squeezing carbon nanotube-based flow inside a rotating stretching channel,” Mathematical Problems in Engineering, vol. 2019, Article ID 9890626, 18 pages, 2019.
View at: Publisher Site | Google Scholar
M. Dehghan, B. N. Saray, and M. Lakestani, “Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers–Huxley equation,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1129–1142, 2012.
View at: Publisher Site | Google Scholar
B. N. Saray, “Abel’s integral operator: sparse representation based on multiwavelets,” BIT Numerical Mathematics, vol. 61, no. 2, pp. 587–606, 2021.
View at: Publisher Site | Google Scholar
B. K. Alpert, “A class of bases in for the sparse representation of integral operators,” SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 246–262, 1993.
View at: Google Scholar
B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, “Wavelet-like bases for the fast solution of second-kind integral equations,” SIAM Journal on Scientific Computing, vol. 14, no. 1, pp. 159–184, 1993.
View at: Publisher Site | Google Scholar
S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, Academic Press, 2008.
B. N. Saray, M. Lakestani, and M. Dehghan, “On the sparse multiscale representation of 2-D Burgers equations by an efficient algorithm based on multiwavelets,” Numerical Methods for Partial Differential Equations, vol. 39, no. 3, pp. 1938–1961, 2023.
View at: Publisher Site | Google Scholar
A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
M. Asadzadeh and B. N. Saray, “On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem,” BIT Numerical Mathematics, vol. 62, no. 4, pp. 1383–1416, 2022.
View at: Publisher Site | Google Scholar
Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual method for solving nonsymmetric linear systems,” SIAM Journal on Scientific and Statistical Computing, vol. 7, no. 3, pp. 856–869, 1986.
View at: Publisher Site | Google Scholar
M. Dehghan, M. Abbaszadeh, and A. Mohebbi, “An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations,” Engineering Analysis with Boundary Elements, vol. 50, pp. 412–434, 2015.
View at: Publisher Site | Google Scholar
A. Mohebbi, M. Abbaszadeh, and M. Dehghan, “High-order difference scheme for the solution of linear time fractional Klein–Gordon equations,” Numerical Methods for Partial Differential Equations, vol. 30, no. 4, pp. 1234–1253, 2014.
View at: Publisher Site | Google Scholar
R. M. Ganji, H. Jafari, M. Kgarose, and A. Mohammadi, “Numerical solutions of time-fractional Klein–Gordon equations by clique polynomials,” Alexandria Engineering Journal, vol. 60, no. 5, pp. 4563–4571, 2021.
View at: Publisher Site | Google Scholar
J. Rashidinia and R. Mohammadi, “Tension spline approach for the numerical solution of nonlinear Klein–Gordon equation,” Computer Physics Communications, vol. 181, no. 1, pp. 78–91, 2010.
View at: Publisher Site | Google Scholar
M. Fardi, “A kernel-based method for solving the time-fractional diffusion equation,” Numerical Methods for Partial Differential Equations, vol. 39, no. 3, pp. 2719–2733, 2023.
View at: Publisher Site | Google Scholar
S. Mohammadi, M. Ghasemi, and M. Fardi, “A fast fourier spectral exponential time-differencing method for solving the time-fractional mobile–immobile advection–dispersion equation,” Computational and Applied Mathematics, vol. 41, no. 6, Article ID 264, 2022.
View at: Publisher Site | Google Scholar
M. Fardi and M. Ghasemi, “A numerical solution strategy based on error analysis for time-fractional mobile/immobile transport model,” Soft Computing, vol. 25, no. 16, pp. 11307–11331, 2021.
View at: Publisher Site | Google Scholar
Z. Bouazza, M. S. Souid, and H. Günerhan, “Multiterm boundary value problem of Caputo fractional differential equations of variable order,” Advances in Difference Equations, vol. 2021, Article ID 400, 2021.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2023 Haifa Bin Jebreen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

124

Downloads

177

Citations