Homotopy perturbation method for solving Caputo‐type fractional‐order Volterra‐Fredholm integro‐differential equations

Das, P. C.; Rana, Subrata; Ramos, Higinio

doi:10.1002/cmm4.1047

Cited by 62 publications

(20 citation statements)

References 16 publications

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“…There are other ways to find an approximation of the solution based on perturbation theory, which can be seen in previous studies. [14][15][16][17] It turns out that the smoothness of the initial data and the structure of the a priori meshes play important roles on the error estimates and convergence of the numerical solutions for SPPs; see, for example, Das and Natesan, 18 Chandru et al 19,20 and Govindarao et al 21 Uniformly convergent numerical methods for the SPPs with delay and advance terms can be also observed in Bansal et al 22,23 on a priori fixed meshes. This mesh structure has been also adaptively improved by Chakravarthy and Kumar 24 by a new strategy.…”

Section: Introductionmentioning

confidence: 93%

A graded mesh refinement approach for boundary layer originated singularly perturbed time‐delayed parabolic convection diffusion problems

Kumar

Chakravarthy

Das

et al. 2021

Math Methods in App Sciences

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In this work, we consider a graded mesh refinement algorithm for solving time-delayed parabolic partial differential equations with a small diffusion parameter. The presence of this parameter leads to boundary layer phenomena. These problems are also known as singularly perturbed problems. For these problems, it is well-known that one cannot achieve a convergent solution to maintain the boundary layer dynamics, on a fixed number of uniform meshes irrespective of the arbitrary magnitude of perturbation parameter. Here, we consider an adaptive graded mesh generation algorithm, which is based on an entropy function in conjunction with the classical difference schemes, to resolve the layer behavior. The advantage of the present algorithm is that it does not require to have any information about the location of the layer. Several examples are presented to show the high performance of the proposed algorithm.

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Section: Introductionmentioning

confidence: 93%

A graded mesh refinement approach for boundary layer originated singularly perturbed time‐delayed parabolic convection diffusion problems

Kumar

Chakravarthy

Das

et al. 2021

Math Methods in App Sciences

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“…There are several notions of fractional derivatives existed in the literature, out of which the Riemann‐Liouville 5,6 and Caputo fractional derivatives 6–8 are mainly popular in recent days. A coupled system of Riemann‐Liouville type fractional differential equations are studied in Jleli and Samet 9 to obtain a sufficient condition for existence of positive solutions.…”

Section: Introductionmentioning

confidence: 99%

Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis

Das

Rana

2021

Math Methods in App Sciences

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In this research, we study a weakly singular Volterra integro differential equation with Caputo‐type fractional derivative. First, we derive a sufficient condition for the existence and uniqueness of the solution of this problem based on the maximum norm. It is observed that the condition depends on the domain of definition of the problem. Thereafter, we show that this condition will be independent of the domain of definition based on an equivalent weighted maximum norm. In addition, we have also provided a procedure to extend the existence and uniqueness of the solution in its domain of definition by partitioning it. We also derive a sufficient condition under which the model problem will provide an analytic solution. Next, we introduce an operator‐based parameterized method to generate an approximate solution of this problem. Convergence analysis of this approach is established here. Next, we have optimized this solution based on least square method. For this, residual minimization is used to obtain the optimal values of the auxiliary parameter. In addition, we have also provided an error bound based on this technique. Several numerical examples are produced to clarify the effective behavior of the convergence of the present method. Comparison of the standard method and optimized method based on residual minimization signify the better accuracy of modified one. In addition, we also consider an equivalent form of weakly singular integro differential equation of Heat transfer problem to show the high effectiveness of the present approach.

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“…Homotopy perturbation method is one of the most useful semi-analytical approach for solving fractional order IDEs. Das et al (2019aDas et al ( , 2019b) used this method to analyze the IDEs involving fractional order derivative of Caputo type. Collocation method is used in Rawashdeh ( 2006) to solve such model numerically.…”

Section: Introductionmentioning

confidence: 99%

Numerical Analysis of Volterra Integro-differential Equations with Caputo Fractional Derivative

Santra

Mohapatra

2021

Iran J Sci Technol Trans Sci

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This article deals with a fully discretized numerical scheme for solving fractional order Volterra integro-differential equations involving Caputo fractional derivative. Such problem exhibits a mild singularity at the initial time t ¼ 0. To approximate the solution, the classical L1 scheme is introduced on a uniform mesh. For the integral part, the composite trapezoidal approximation is used. It is shown that the approximate solution converges to the exact solution. The error analysis is carried out. Due to presence of weak singularity at the initial time, we obtain the rate of convergence is of order OðsÞ on any subdomain away from the origin whereas it is of order Oðs a Þ over the entire domain. Finally, we present a couple of examples to show the efficiency and the accuracy of the numerical scheme.

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Homotopy perturbation method for solving Caputo‐type fractional‐order Volterra‐Fredholm integro‐differential equations

Cited by 62 publications

References 16 publications

A graded mesh refinement approach for boundary layer originated singularly perturbed time‐delayed parabolic convection diffusion problems

A graded mesh refinement approach for boundary layer originated singularly perturbed time‐delayed parabolic convection diffusion problems

Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis

Numerical Analysis of Volterra Integro-differential Equations with Caputo Fractional Derivative

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