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

Abstract

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.

Fractional View Analysis of Third Order Kortewege-De Vries Equations, Using a New Analytical Technique

“…The KdV-Burgers equation defines the waves on lower water surfaces. The strength of fractional KdV equation is the non-local property [11][12][13][14][15][16][17][18][19][20][21]. For a higher order Korteweg-de Vries equation, which is a natural extension of the Korteweg-de Vries equation written in a bilinear form, a Bcklund conversion in bilinear forms is provided.…”

Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations

“…There is a clear advantage of the suggested method that it works without any use of Adomian polynomials, as required by the Adomian decomposition method. Results of the analysis show that the suggested method produces the solution in a series of fast convergence that can result in a closed solution [25][26][27][28][29][30].…”

Numerical Optics Soliton Solution of the Nonlinear Schrödinger Equation Using the Laplace and the Modified Laplace Decomposition Method

“…in which 0 PQ  gives the de-focusing case,  is a soliton velocity, Last recent decades, the methods of decomposing have emerged as a powerful technique and as a subject of extensive analytical and numerical studies for large and general class of linear and nonlinear ordinary differential equations (ODE's) as well as partial differential equations (PDE's), fractional differential equations, algebraic, integro-differential, differential-delay equations [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. More precisely, the Adomian decomposition method is knowingly efficient in solving initial-value or boundary value problems without unphysical restrictive assumptions such as linearization, perturbation and so forth.…”

“…Laplace Decomposition Method (LDM) was introduced by Khuri [11,12] and has been successfully utilized for obtaining solutions of differential equations [6,7,9,14,17,[26][27][28][29][30][31][32][33][34] and the NLSE of our interest. As, for instance, a recent study by Gaxiola [26] who applied the Laplace-Adomian decomposition method to a NLSlike equation, namely the Kundu-Eckhaus equation, and the accuracy as well as the efficiency of the method is proved via examples, as for the nonlinear Schrodinger equation with harmonic oscillator the method of Laplace-Adomian was utilized in a comparison with another semi-analytical method to obtain approximate analytical solutions by Jaradat et al [28] .…”