Department of Mathematics

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Author: search.filters.author.Santra, SudarshanSubject: search.filters.subject.Caputo fractional derivative

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    Numerical analysis of volterra integro-differential equations with caputo fractional derivative
    (Springer, 2021-07) Santra, Sudarshan
    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 . 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 on any subdomain away from the origin whereas it is of order 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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    A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra type
    (Elsevier, 2022-01) Santra, Sudarshan
    The main purpose of this work is to study the numerical solution of a time fractional partial integro-differential equation of Volterra type, where the time derivative is defined in Caputo sense. Our method is a combination of the classical L1 scheme for temporal derivative, the general second order central difference approximation for spatial derivative and the repeated quadrature rule for integral part. The error analysis is carried out and it is shown that the approximate solution converges to the exact solution. Several examples are given in support of the theoretical findings. In addition, we have shown that the order of convergence is more high on any subdomain away from the origin compared to the entire domain.
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    Adomian decomposition and homotopy perturbation method for the solution of time fractional partial integro-differential equations
    (Springer, 2021-07) Santra, Sudarshan
    This article deals with two different methods to solve a time fractional partial integro-differential equation. The fractional derivatives are defined here in Caputo sense. The model problem is solved using the Adomian decomposition method and homotopy perturbation method. Moreover, this paper proves the convergence analysis of the solution based on the present methods. Numerical evidences are illustrated in support of the theoretical analysis.
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    A novel approach for solving multi-term time fractional Volterra–Fredholm partial integro-differential equations
    (Springer, 2021-12) Santra, Sudarshan
    This article deals with an efficient numerical technique to solve a class of multi-term time fractional Volterra–Fredholm partial integro-differential equations of first kind. The fractional derivatives are defined in Caputo sense. The Adomian decomposition method is used to construct the scheme. For simplicity of the analysis, the model problem is converted into a multi-term time fractional Volterra–Fredholm partial integro-differential equation of second kind. In addition, the convergence analysis and the condition for existence and uniqueness of the solution are provided. Several numerical examples are illustrated in support of the theoretical analysis.
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    Analytical and numerical solution for the time fractional black-scholes model under jump-diffusion
    (Springer, 2023-04) Santra, Sudarshan
    In this work, we study the numerical solution for time fractional Black-Scholes model under jump-diffusion involving a Caputo differential operator. For simplicity of the analysis, the model problem is converted into a time fractional partial integro-differential equation with a Fredholm integral operator. The L1 discretization is introduced on a graded mesh to approximate the temporal derivative. A second order central difference scheme is used to replace the spatial derivatives and the composite trapezoidal approximation is employed to discretize the integral part. The stability results for the proposed numerical scheme are derived with a sharp error estimation. A rigorous analysis proves that the optimal rate of convergence is obtained for a suitable choice of the grading parameter. Further, we introduce the Adomian decomposition method to find out an analytical approximate solution of the given model and the results are compared with the numerical solutions. The main advantage of the fully discretized numerical method is that it not only resolves the initial singularity occurred due to the presence of the fractional operator, but it also gives a higher rate of convergence compared to the uniform mesh. On the other hand, the Adomian decomposition method gives the analytical solution as well as a numerical approximation of the solution which does not involve any mesh discretization. Furthermore, the method does not require a large amount of computer memory and is free of rounding errors. Some experiments are performed for both methods and it is shown that the results agree well with the theoretical findings. In addition, the proposed schemes are investigated on numerous European option pricing jump-diffusion models such as Merton’s jump-diffusion and Kou’s jump-diffusion for both European call and put options.