Browsing by Author "Santra, Sudarshan"

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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.
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.
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.
Analysis of a higher-order scheme for multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels
(Springer, 2024-09) Santra, Sudarshan
This work is focused on developing a hybrid numerical method that combines a higher-order finite difference method and multi-dimensional Hermite wavelets to address two-dimensional multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels having bounded and unbounded time derivatives at the initial time . Specifically, the multi-term fractional operators are discretized using a higher-order approximation designed by employing different interpolation schemes based on linear, quadratic, and cubic interpolation leading to accuracy on a suitably chosen nonuniform mesh and accuracy on a uniformly distributed mesh. The weakly singular integral operators are approximated by a modified numerical quadrature, which is a combination of the composite trapezoidal approximation and the midpoint rule. The effects of the exponents of the weakly singular kernels over fractional orders are analyzed in terms of accuracy over uniform and nonuniform meshes for the solution having both bounded and unbounded time derivatives. The stability of the proposed semi-discrete scheme is derived based on -norm for uniformly distributed temporal mesh. Further, we employ the uniformly distributed collocation points in spatial directions to estimate the tensor-based wavelet coefficients. Moreover, the convergence analysis of the fully discrete scheme is carried out based on -norm leading to accuracy on a uniform mesh. It also highlights the higher-order accuracy over nonuniform mesh. Additionally, we discuss the convergence analysis of the proposed scheme in the context of the multi-term time-fractional diffusion equations involving time singularity demonstrating a accuracy on a nonuniform mesh with suitably chosen grading parameter. Note that the scheme reduces to accuracy on a uniform mesh. Several tests are performed on numerous examples in - and -norm to show the efficiency of the proposed method. Further, the solutions’ nature and accuracy in terms of absolute point-wise error are illustrated through several isosurface plots for different regularities of the exact solution. These experiments confirm the theoretical accuracy and guarantee the convergence of approximations to the functions having time singularity, and the higher-order accuracy for a suitably chosen nonuniform mesh.
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.
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.
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.
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.
Enhancing accuracy with an adaptive discretization for the non-local integro-partial differential equations involving initial time singularities
(Elsevier, 2025-08) Santra, Sudarshan
This work aims to construct an efficient and highly accurate numerical method to address the time singularity at involved in a class of time-fractional parabolic integro-partial differential equations in one and two dimensions. The L2- scheme is used to discretize the time-fractional operator, whereas a modified version of the composite trapezoidal approximation is employed to discretize the Volterra operator in time. Subsequently, it helps to convert the proposed model into a second-order boundary value problem in a semi-discrete form. The multi-dimensional Haar wavelets are then used for grid adaptation and efficient computations for the two-dimensional problem, whereas the standard second-order approximations are employed to approximate the spatial derivatives for the one-dimensional case. The stability analysis is carried out on an adaptive mesh in time. The convergence analysis leads to accurate solution in the space-time domain for the one-dimensional problem having time singularity based on the norm for a suitable choice of the grading parameter. Furthermore, it provides accurate solution for the two-dimensional problem having unbounded time derivative at . The analysis also highlights a higher order accuracy for a sufficiently smooth solution resides in even if the mesh is discretized uniformly. The truncation error estimates for the time-fractional operator, integral operator, and spatial derivatives are presented. In addition, we have examined the impact of various parameters on the robustness and accuracy of the proposed method. Numerous tests are performed on several examples in support of the theoretical analysis. The advancement of the proposed methodology is demonstrated through the application of the time-fractional Fokker-Planck equation and the fractional-order viscoelastic dynamics having weakly singular kernels. It also confirms the superiority of the proposed method compared with existing approaches available in the literature.
Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis
(Elsevier, 2023-11) Santra, Sudarshan
This work deals with a higher order numerical approximation for analyzing a class of multi-term time fractional partial integro-differential equations involving Volterra integral operators. The solutions to these problems have a mild singularity at the initial time, due to which an initial layer appears, which becomes more sharper as the highest order time fractional derivative decreases. This behaviour reduces the rate of convergence by standard approaches. We start the present work by considering the existence and uniqueness of a class of generalized partial integro-differential equations and then, present the L1 discretization on a graded mesh in time which is adapted towards the initial time level. This discretization leads to a higher order accuracy than the solutions obtained on a time uniform mesh. The convergence analysis corresponding to the Volterra integral operator is nontrivial as it uses a repeated quadrature rule. This analysis can also be extended for weakly singular kernels. The stability analysis of the present scheme with a sharp error estimation is also provided. The analysis with extensive experiments shows that a higher rate of accuracy can be attained for several suitable choices of the grading parameters for solving several classes of time fractional integro-differential equations.
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.
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.
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.
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.
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"/>
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.
Physics-informed fractional machine intelligence and space–time wavelet frameworks for non-local integro-partial differential equations involving weak singularities
(Elsevier, 2026-01) Santra, Sudarshan
This paper presents a space–time multi-dimensional wavelet framework and a physics-informed fractional machine intelligence (PI-fMI) model to address the weak singularity involved in time-fractional integro-partial differential equations with mixed Volterra–Fredholm operators. Conventional machine learning approaches often struggle with weak initial singularities; however, our proposed approach overcomes this challenge through two complementary strategies in the context of fractional-order integro-differential equations. First, a wavelet-based numerical scheme is employed that utilizes the multi-resolution analysis with the collocation method to compute the wavelet coefficients, ensuring convergence for fractional-order integro-differential problems with sufficiently smooth solutions. Second, we introduce a PI-fMI model for problems that exhibit unbounded temporal derivatives at , which incorporates the discretization for fractional operators, a combination of the repeated quadrature rule, and automatic differentiation to handle integral operators that contain diffusion terms. Theoretical and numerical analyses demonstrate that the proposed approach successfully resolves the initial weak singularities where the traditional Haar wavelets fail to address such issues. Furthermore, the convergence of the PI-fMI model is analyzed for problems with nonlinear source terms, demonstrating its effectiveness under suitable hyperparameter choices. Theoretical findings are validated through extensive numerical experiments on several test problems exhibiting bounded and unbounded temporal derivatives at .
Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations
(Elsevier, 2025-01) Santra, Sudarshan
An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at . In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.