Numerical Solutions of Fractional Systems of Two-Point BVPs by Using the Iterative Reproducing Kernel Algorithm

Altawallbeh, Zuhier; Al‐Smadi, Mohammed; Komashynska, Iryna; Ateiwi, Ali Mahmud

doi:10.1007/s11253-018-1526-8

Cited by 47 publications

(27 citation statements)

References 18 publications

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“…However, many numerical and analytical techniques have been employed recently for solving stiff systems of ordinary differential equations including the homotopy perturbation method [3], the block method [4], the multistep method [5], and the variational iteration method [6]. Ex-amples of another mathematical models and effective numerical solutions can be found in [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

et al. 2019

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A powerful analytical approach, namely the fractional residual power series method (FRPS), is applied successfully in this work to solving a class of fractional stiff systems. The methodology of the FRPS method gets a Maclaurin expansion of the solution in rapidly convergent form and apparent sequences based on the Caputo sense without any restriction hypothesis. This approach is tested on a fractional stiff system with nonlinearity ranging. Meanwhile, stability and convergence study are presented in the domain of interest. Illustrative examples justify that the proposed method is analytically effective and convenient, and it can be implemented in a large number of engineering problems. A numerical comparison for the experimental data with another well-known method, the reproducing kernel method, is given. The graphical consequences illuminate the simplicity and reliability of the FRPS method in the determination of the RPS solutions consistently.

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Section: Introductionmentioning

confidence: 99%

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

et al. 2019

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“…In this section, we have given some basic definitions and theorems regarding the reproducing-kernel spaces and the generalized power series representations. For more details about these definitions and properties, one can refer to [22][23][24][25][26][27]. Throughout the current paper,…”

Section: Basic Concepts and Fundamentalsmentioning

confidence: 99%

“…and D α a u(x) = u (n) (x) for α = n ∈ N. The reproducing-kernel approach has been developed as an efficacious numericanalytic method in treating different type of ordinary and partial differential, integral, integrodifferential equations with singularity, fuzziness, nonlocal, and non-classical constraint conditions [22][23][24][25][26][27]. Recently, the RKM has been improved and successfully applied in obtaining approximations of solutions for many initial and boundary problems that appear in natural sciences and engineering.…”

Section: Introductionmentioning

confidence: 99%

Two computational approaches for solving a fractional obstacle system in Hilbert space

et al. 2019

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The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0 < α ≤ 1, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

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“…Unlike the classical calculus, which has unique definitions and clear geometrical and physical interpretations, there are numerous definitions of the fractional operations. Riemann-Liouville, Riesz, Grünwald-Letnikov, and Caputo are some examples of these definitions [5][6][7][8]. Recently, FDEs have attracted the attention of numerous researchers for its considerable importance in many scientific applications, including fluid dynamics, signal processing, viscoelasticity, bioengineering, finance, Hamiltonian chaos, and vibrations [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

“…Riemann-Liouville, Riesz, Grünwald-Letnikov, and Caputo are some examples of these definitions [5][6][7][8]. Recently, FDEs have attracted the attention of numerous researchers for its considerable importance in many scientific applications, including fluid dynamics, signal processing, viscoelasticity, bioengineering, finance, Hamiltonian chaos, and vibrations [1][2][3][4][5][6]. In this light, there exists no classic, precise method that yields an analytical solution in a closed-form for these models; therefore, approximate and numerical methods have been developed to handle such FDEs.…”

Section: Introductionmentioning

confidence: 99%

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

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Featured Application: Fractional differential equations play a significant role in modeling certain dynamical systems arising in many fields of applied sciences and engineering. In this paper, the authors develop an attractive analytic-numeric technique, residual power series (RPS) method, for solving fractional Bagley-Torvik equations with a source term involving Caputo fractional derivative. In regard to its simplicity, the method can be applicable to a wide class of fractional partial differential equations, fractional fuzzy differential equations, fractional oscillator equations, and so on.Abstract: Numerical simulation of physical issues is often performed by nonlinear modeling, which typically involves solving a set of concurrent fractional differential equations through effective approximate methods. In this paper, an analytic-numeric simulation technique, called residual power series (RPS), is proposed in obtaining the numerical solution a class of fractional Bagley-Torvik problems (FBTP) arising in a Newtonian fluid. This approach optimizes the solutions by minimizing the residual error functions that can be directly applied to generate fractional PS with a rapidly convergent rate. The RPS description is presented in detail to approximate the solution of FBTPs by highlighting all the steps necessary to implement the algorithm in addressing some test problems. The results indicate that the RPS algorithm is reliable and suitable in solving a wide range of fractional differential equations applying in physics and engineering.

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Numerical Solutions of Fractional Systems of Two-Point BVPs by Using the Iterative Reproducing Kernel Algorithm

Cited by 47 publications

References 18 publications

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

Two computational approaches for solving a fractional obstacle system in Hilbert space

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

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