Comparison between the Laplace Decomposition Method and Adomian Decomposition in Time-Space Fractional Nonlinear Fractional Differential Equations

Mohamed, Mohamed Z.

doi:10.4236/am.2018.94032

Cited by 28 publications

(21 citation statements)

References 13 publications

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“…As compared to other analytical techniques, LADM have less parameters, therefore LADM is an efficient technique, requiring no discretization and linearization [ 22 ]. A comparison between the LADM and ADM for the analysis of FDEs is given in [ 23 ]. The Kundu–Eckhaus equation deals with quantum field theory, and the analytical solution of this nonlinear PDEs has been studied in [ 24 ] using LADM.…”

Section: Introductionmentioning

confidence: 99%

Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

Shah

Khan

Arif

et al. 2019

Entropy

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In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.

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Section: Introductionmentioning

confidence: 99%

Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

Shah

Khan

Arif

et al. 2019

Entropy

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“…Non-linear Coupled PDE's and non-linear Blasius flow equation using Laplace decompostion method [13,14]. A comparison between the LADM and ADM for the analysis of FPDEs is discussed in [15]. The Kundu-Eckhaus Equation deals in the quantum field theory, and the analytical solution of this nonlinear PDEs has been derived in [10] using LADM.…”

Section: Introductionmentioning

confidence: 99%

An Analytical Technique to Solve the System of Nonlinear Fractional Partial Differential Equations

2019

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The Kortweg–de Vries equations play an important role to model different physical phenomena in nature. In this research article, we have investigated the analytical solution to system of nonlinear fractional Kortweg–de Vries, partial differential equations. The Caputo operator is used to define fractional derivatives. Some illustrative examples are considered to check the validity and accuracy of the proposed method. The obtained results have shown the best agreement with the exact solution for the problems. The solution graphs are in full support to confirm the authenticity of the present method.

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“…Many physical phenomena that are modeled by partial differential equations and fractional‐order partial differential equations are solved by using the LADM. The numerical solution of the fractional‐order Whitham‐Broer‐Kaup equations are presented in Ali et al; the solution of linear and nonlinear fractional‐order partial differential equations are successfully found in Ahmed et al; the approximate solution of a nonlinear fractional‐order Volterra and Fredholm integro‐differential equations was discussed in Irfan et al; a system of delay differential equations was successfully described and investigated in Yousef and Ismail; the solution of some well‐known diffusion equations was given in Jafari et al; the application of the proposed method for other nonlinear partial differential equations can be found in Mohamed and Elzaki, and so on.…”

Section: Introductionmentioning

confidence: 99%

Some analytical and numerical investigation of a family of fractional‐order Helmholtz equations in two space dimensions

Srivástava

Shah

Khan

et al. 2019

Math Methods in App Sciences

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In this article, we aim at solving a family of two-dimensional fractional-order Helmholtz equations by using the Laplace-Adomian Decomposition Method (LADM). The fractional-order derivatives, which we use in this investigation, follows the Liouville-Caputo definition. Our results based upon the LADM are obtained in series form that helps us in analyzing the analytical solutions of the fractional-order Helmholtz equations considered here. For illustration and verification of the analytical procedure using the LADM, several numerical examples and graphical representations are presented for the analytical solution of the fractional-order Helmholtz equations. The mathematical analytic procedure, which we have used here, has shown that the LADM is a fairly accurate and computable method for the solution of problems involving fractional-order Helmholtz equations in two dimensions. In an analogous manner, one can apply the LADM for finding the analytical solution of other classes of fractional-order partial differential equations. KEYWORDS Adomian decomposition method (ADM), fractional calculus, fractional-order Helmholtz equations, finite-difference method (FDM), Laplace-Adomian decomposition method (LADM), Liouville-Caputo derivative operator, Mittag-Leffler functions, Riemann-Liouville derivative operator MSC CLASSIFICATION 35J15; 35J70; 58J10; 58J20 | INTRODUCTION AND MOTIVATIONFractional calculus plays an important rôle in applied mathematics because of its various applications in modeling of many different physical phenomena occurring in science and engineering. The concept of a derivative, D α { f (x)} with nonintegerorder α, has improved the early development of the ordinary derivative with nonnegative integer values of the order α. In fact, in the development of fractional calculus, many noted mathematical scientists, such as Euler, Riemann, Liouville, Weyl, Leibniz, Fourier, Bernoulli, l'Hôpital, Wallis, and others, made remarkable contributions to this subject and other related research areas. In recent years, researchers in the mathematical, physical, and engineering sciences have made significant contributions toward the theory and applications of fractional calculus. Fractional calculus has many applications in the fields of (for example) viscoelasticity,

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Comparison between the Laplace Decomposition Method and Adomian Decomposition in Time-Space Fractional Nonlinear Fractional Differential Equations

Cited by 28 publications

References 13 publications

Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

Application of Laplace–Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations

An Analytical Technique to Solve the System of Nonlinear Fractional Partial Differential Equations

Some analytical and numerical investigation of a family of fractional‐order Helmholtz equations in two space dimensions

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