Department of Mathematics

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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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    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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    An adaptive mesh based computational approach to the option price and their greeks in time fractional black–scholes framework
    (Springer, 2025-02) Santra, Sudarshan
    This article deals with an efficient numerical method for solving the time fractional Black–Scholes equation governing the European option pricing model and their Greeks. The Caputo fractional derivative involved in time results a mild singularity and forms a layer near the initial time. For discretization, a graded mesh is introduced in the temporal direction, and in space, a uniform mesh is constructed. The L1 scheme is used to discretize the time fractional derivative, while the second-order finite difference approximations are used for the spatial derivatives. The proposed approach effectively resolves the initial layer with a graded mesh in time, achieving higher temporal accuracy of . It provides valuable insights into the error bounds through stability and convergence analysis and captures the behavior of option Greeks, highlighting the impact of fractional derivatives. Compared to uniform mesh-based methods and other existing approaches, it demonstrates superior accuracy and efficiency for time-fractional Black–Scholes equations, ensuring space-time higher-order accuracy. Some numerical results on the solution and their Greeks prove the theoretical analysis. The proposed scheme is applied to European option pricing models governed by the time fractional Black–Scholes equation to examine the impact of the fractional derivative on option pricing.
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    An efficient hybrid numerical approach for time-fractional sub-diffusion equations with multi-singularities
    (Springer, 2025-06) Santra, Sudarshan
    The main focus of this work is to develop a hybrid numerical method based on the L1 scheme and the multi-dimensional Hermite wavelets. We discuss the stability and convergence analysis on a newly designed time-graded mesh to address a class of time-fractional delay partial differential equations involving multi-singularities. In the context of multi-singularities, there are significant challenges for higher-dimensional problems, and the available analytical framework exhibits substantial limitations. Addressing these challenges requires innovative approaches that can effectively navigate the increased complexity of higher-dimensional problems while maintaining analytical rigor and computational efficiency. We use the L1 scheme to convert the proposed problem into a semi-discrete form. The stability and convergence of the temporal semi-discretization on the newly constructed graded mesh are analyzed based on -norm that leads to temporal rate of accuracy for a suitably chosen grading parameter. The strength of the newly constructed mesh is that it provides a more robust and accurate approach to address multi-singularities and has less computational cost to achieve the desired accuracy compared to other meshes available in the literature. The multi-dimensional Hermite wavelet approximation is taken into account to solve the semi-discrete problem and we use uniformly distributed collocation points in the spatial direction to estimate the unknown wavelet coefficients. Further, the convergence analysis of the proposed hybrid numerical approximation leads to rate of accuracy over the space-time domain based on -norm for a suitable choice of the grading parameter. In particular, the performance of the hybrid numerical approach is verified through numerous complex problems involving multiple delay parameters.