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An analytical solution with existence and uniqueness conditions for fractional integro-differential equations

International Journal of Modeling, Simulation, and Scientific Computing 2020年第5期11卷 147-169页

作者： Pratibha Verma Manoj Kumar Department of Mathematics Motilal Nehru National Institute of Technology AllahabadPrayagraj-211004Uttar PradeshIndia

This study aims to apply the two-step Adomian decomposition method(TSADM)to find an analytical solution of integro-differential equations for fractional order without discretization/approximation with less number of iterations and reduce the computational efforts.Moreover,we have established the results for the existence and uniqueness of a solution with the help of some fixed point theorems and the Banach contraction principle.Furthermore,the method is demonstrated on different test examples arising in real life situations.It is concluded that the TSADM provides exact solution of the fractional integro-differential equations in one iteration.At the same time,the other existing methods furnish an approximate solution and require lots of computation to solve the problem applying discretization/approximation on fractional operators.

关键词： Caputo fractional derivative two-step Adomian decomposition method integro-differential equation fixed point theorem

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Ulam-Hyers stability and analytical approach for m-dimensional Caputo space-time variable fractional order advection-dispersion equation

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International Journal of Modeling, Simulation, and Scientific Computing 2022年第1期13卷 131-174页

作者： Pratibha Verma Manoj Kumar Anand Shukla Department of Mathematics Motilal Nehru National Institute of Technology Allahabad Prayagraj–211004 Uttar PradeshIndia†Department of Mathematics Wollega UniversityNekemteEthiopia

This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration.Moreover,with the help of fixed point theory,we study the existence and uniqueness conditions for the positive solution and prove some new results.Also,obtain the Ulam–Hyers stabilities for the proposed problem.two gen-eralized examples are considered to show the method’s applicability and compared with other existing numerical methods.The present method performs exceptionally well in terms of efficiency and simplicity.Further,we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.

关键词： Fixed point theorems space-time variable Caputo’s fractional operators advection-dispersion equation Ulam-Hyers stability two-step Adomian decomposition method

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An analytical solution of multi-dimensional space fractional diffusion equations with variable coefficients

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International Journal of Modeling, Simulation, and Scientific Computing 2021年第1期12卷 229-255页

作者： Pratibha Verma Manoj Kumar Department of Mathematics Motilal Nehru National Institute of Technology Allahabad Prayagraj 211004 Uttar PradeshIndia

In this paper,we have considered the multi-dimensional space fractional diffusion equations with variable coefficients.The fractional operators(derivative/integral)are used based on the Caputo definition.This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method(TSADM).Moreover,new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem.We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions.Finally,the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods.The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM.There are no restrictions imposed on the problems for diffusion coefficients,and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.

关键词： Caputo fractional operators space fractional diffusion equations Riesz derivative two-step Adomian decomposition method fixed point theorem

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