Modification of Laplace Adomian Decomposition Method for Solving Nonlinear Volterra Integral and Integro-differential Equations based on Newton Raphson Formula

Rani, Dimple; Mishra, Vinod

doi:10.29020/nybg.ejpam.v11i1.2645

Cited by 19 publications

(16 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the literature, there are many definitions of fractional derivative, the most popular definitions are of Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio, Atangana-Baleanu, Riesz, Hilfer, among others [8][9][10]. Recently, several numerical methods have been proposed to obtain approximate solutions of fractional ordinary differential equations and fractional partial differential equations, such as the fractional sub-equation method [11,12], the Adomian decomposition method [13][14][15], the Homotopy perturbation method [16,17], the variational iteration method [18][19][20][21], homotopy perturbation transform method [22,23], and so on.…”

Section: Introductionmentioning

confidence: 99%

The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative

Ghanbari

Gómez-Aguilar

2019

Rev. Mex. Fís.

View full text Add to dashboard Cite

In this paper, the generalized exponential rational function method (GERFM) and the extended sinh-Gordon equation expansion method (ShGEEM) are used to construct exact solutions of the perturbed β-conformable-time Radhakrishnan-Kundu-Lakshmanan (RKL) equation. This model governs soliton propagation dynamics through a polarization-preserving fiber. Fractional derivatives are described in the β-conformable sense. As a result, we get new form of solitary traveling wave solutions for this model including novel soliton, traveling waves and kink-type solutions with complex structures. Physical interpretations of some extracted solutions are also included through taking suitable values of parameters and derivative order in them. It is proved that these methods are powerful, efficient, and can be fruitfully implemented to establish new solutions of nonlinear conformable-time partial differential equations applied in mathematical physics.

show abstract

Section: Introductionmentioning

confidence: 99%

The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative

Ghanbari

Gómez-Aguilar

2019

Rev. Mex. Fís.

View full text Add to dashboard Cite

show abstract

“…The ADM correlated with several integral transforms, such as Laplace, modified Laplace, Mohand, Aboodh, Elzaki and many more. Recently, modified Laplace ADM [35] for solving nonlinear Volterra integral and integro-differential equations based on the Newton-Raphson formula, Discrete ADM [36] used for solving time-fractional Navier-Stokes equation, Laplace ADM [37] for finding the numerical solution of a fractional order epidemic model of a vector-born disease and hence forth.…”

Section: Introductionmentioning

confidence: 99%

A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

et al. 2021

View full text Add to dashboard Cite

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

show abstract

“…Accurate models of systems under consideration can be obtained using fractional differential and integrodifferential equations [11]. Many remarkable works on fractional calculus are available in literature for the approxi-mation of the solution fractional differential or integrodifferential equations, for example, the sinc-collocation method [12], Legendre collocation method [13], Laguerre polynomials [14], Adomian decomposition method [15], Variational iteration method [16][17][18], and their references.…”

Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

Qiang

Kamran

Mahboob

et al. 2020

Journal of Function Spaces

View full text Add to dashboard Cite

Laplace transform is a powerful tool for solving differential and integrodifferential equations in engineering sciences. The use of Laplace transform for the solution of differential or integrodifferential equations sometimes leads to the solutions in the Laplace domain that cannot be inverted to the real domain by analytic methods. Therefore, we need numerical methods to invert the solution to the real domain. In this work, we construct numerical schemes based on Laplace transform for the solution of fractional-order Volterra integrodifferential equations in the sense of the Atangana-Baleanu Caputo derivative. We propose two numerical methods for approximating the solution of fractional-order linear and nonlinear Volterra integrodifferential equations. In our scheme, the inverse Laplace transform is approximated using a contour integration method and Stehfest method. Some numerical experiments are performed to check the accuracy and efficiency of the methods. The results obtained using these methods are compared.

show abstract

Modification of Laplace Adomian Decomposition Method for Solving Nonlinear Volterra Integral and Integro-differential Equations based on Newton Raphson Formula

Cited by 19 publications

References 8 publications

The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative

The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with β-conformable time derivative

A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

Contact Info

Product

Resources

About