Department of Mathematics

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Author: search.filters.author.Santra, Sudarshan

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    Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity
    (Wiley, 2020-09) Santra, Sudarshan
    A time fractional initial boundary value problem of mixed parabolic–elliptic type is considered. The domain of such problem is divided into two subdomains. A reaction–diffusion parabolic problem is considered on the first domain, and on the second, a convection–diffusion elliptic type problem is considered. Such problem has a mild singularity at the initial time t = 0. The classical L1 scheme is introduced to approximate the temporal derivative, and a second order standard finite difference scheme is used to approximate the spatial derivatives. The domain is discretized with uniform mesh for both directions. It is shown that the order of convergence is more higher away from t = 0 than the order of convergence on the whole domain. To show the efficiency of the scheme, numerical results are provided.
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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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    Numerical treatment of multi-term time fractional nonlinear KdV equations with weakly singular solutions
    (Taylor & Francis, 2021-12) Santra, Sudarshan
    The main aim of this work is to construct an efficient recursive numerical technique for solving multi-term time fractional nonlinear KdV equation. The fractional derivatives are defined in Caputo sense. A modified Laplace decomposition method is introduced to approximate the solution. The Adomian polynomials play an important role to execute such a recursive process. In addition, the mathematical importance and some applications of KdV equation are discussed. The approximate solution obtained by the proposed method can be expressed in the form of an infinite convergent series. The experimental evidences demonstrate the effectiveness of the proposed method.
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    Analysis of a finite difference method based on L1 discretization for solving multi-term fractional differential equation involving weak singularity
    (Wiley, 2022-03) Santra, Sudarshan
    In this article, we consider a multi-term fractional initial value problem which has a weak singularity at the initial time . The fractional derivatives are defined in Caputo sense. Due to such singular behavior, an initial layer occurs near which is sharper for small values of γ1 where γ1 is the highest order among all fractional differential operators. In addition, the analytical properties of the solution are provided. The classical L1 scheme is introduced on a uniform mesh to approximate the fractional derivatives. The error analysis is carried out, and it is shown that the numerical solution converges to the exact solution. Further analysis proves that the scheme is of order over the entire region, but it is of order O(τ) on any subdomain away from the origin. τ denotes the mesh parameter. To show the efficiency of the proposed scheme, this method is tested on several model problems, and the results are in agreement with the theoretical findings.
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    An efficient computational approach for the solution of time-space fractional diffusion equation
    (Taylor & Francis, 2022-06) Santra, Sudarshan
    The main aim of this paper is to construct an efficient recursive algorithm to solve a time-space fractional Poisson’s equation which can be treated as a time-space fractional diffusion equation in two dimensions. The fractional derivatives in both time and space are defined in the Caputo sense. A homotopy perturbation method is introduced to approximate the solution, and a comparison is made between the exact and the approximate solutions. In addition, we present a procedure for solving higher-order fractional Poisson’s equations. In this case, the equation is converted to a system of fractional differential equations in which the order of the time derivatives is less than or equal to one. The convergence analysis is carried out, and an apriori bound of the solution is obtained for the present problem. Numerical examples are provided and the experimental evidence proves the effectiveness of the proposed method.
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    Numerical simulation and convergence analysis for Riemann-Liouville fractional initial value problem involving weak singularity
    (Inder Science, 2023-11) Santra, Sudarshan
    The present work considers a Riemann-Liouville fractional initial value problem (IVP) associated with homogeneous initial condition involving a weak singularity near the origin. Due to presence of initial singularity, an initial layer occurs at t = 0. The L1 scheme is introduced on a uniform mesh to approximate the solution. The convergence analysis shows that the present method is more accurate and produces less error compared to some existing methods on any subdomain away from the origin while the proposed method is comparable over the entire region. Numerical examples and comparison results are provided in order to show the effectiveness of the proposed method.
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    Numerical simulation for time fractional integro partial differential equations arising in viscoelastic dynamical system
    (CRC Press, 2023) Santra, Sudarshan
    The study on fractional calculus gains more attention of many researchers in recent times, due to its immense applicability to define various models, such as viscoelastic damped structure [1], the model due to radiative transfer [2], the theory of linear transport [3], and the mathematical structure due to kinetic energy of gases [4]. A detailed investigation about the application of fractional differential as well as fractional integro-differential equation is available in [5–7]. The general form of a fractional derivative viscoelastic models can be written as: 8.1 https://www.w3.org/1998/Math/MathML" display="block"> X ( t ) + ∑ m = 1 M a m D t α m X ( t ) = E 0 Y ( t ) + ∑ n = 1 N E n D t β n Y ( t ) , https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003328032/39998614-bd30-4270-a56c-d58717d36a18/content/math8_1.tif"/>