Department of Mathematics

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    Analyzing unemployment dynamics: a fractional-order mathematical model
    (Wiley, 2025-03) Mathur, Trilok
    The persistent rise in unemployment rates poses a significant threat to global economic stability. Addressing this challenge effectively requires a deeper understanding of workforce dynamics, particularly through the integration of an individual's employment history into analytical models. This research introduces a fractional mathematical model, leveraging the Caputo fractional derivative and three key variables: the number of skilled unemployed individuals, the number of employed individuals, and the number of available job vacancies. The model's well-posedness and global stability are rigorously established using fixed-point theory. Additionally, the basic reproduction number is analyzed to identify critical factors that facilitate the creation of new job opportunities. Real-world data from India are employed for MATLAB simulations, offering predictions of unemployment trends in the coming years. A graphical analysis highlights the impact of the COVID-19 pandemic on unemployment rates. The model's predictive accuracy is demonstrated through error analysis, showing that fractional-order forecasts achieve less than 5% error, outperforming integer-order models in capturing the nuances of unemployment dynamics. Sensitivity analysis reveals that the employment rate is the most influential parameter; a 40% increase in this rate could lead to 192,200 additional employed individuals. The fractional-order model further exhibits superior performance metrics, including lower root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) values, alongside a higher correlation coefficient ( ). These findings underscore the model's potential to enhance the understanding and mitigation of unemployment challenges.
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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.