In this article we introduce a finite difference approximation for integro-differential operators of Lévy type. We approximate solutions of integro-differential equations, where the second order operator is allowed to degenerate. In the existing literature, the Lévy operator is treated as a zero/first order operator outside of a centered ball of radius $δ$, leading to error estimates of order $ξ(δ)+N(δ)(h+\sqrtτ)$, where $h$ and $τ$ are the spatial and temporal discretization parameters respectively. In these estimates $ξ(δ) \downarrow 0$, but $N(δ)\uparrow \infty$ as $δ\downarrow 0$. In contrast, we treat the integro-differential operator as a second order operator on the whole unit ball. By this method we obtain error estimates of order $(h+τ^k)$ for $k\in \{1/2,1\}$, eliminating the additional errors and the blowing up constants. Moreover, we do not pose any conditions on the Lévy measure.
In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce.
In this paper, we propose a method to approximate the fixed point of an operator in a Banach space. Using biorthogonal systems, this method is applied to build an approximation of the solution of a class of nonlinear partial integro--differential equations. The theoretical findings are illustrated with several numerical examples, confirming the reliability, validity and precision of the proposed method.
We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator's banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form $K(x,y)=K(x-y)$ but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments on problems with or without known analytic solutions and comparisons with other methods.
The aim of the present paper is to introduce a new numerical method for solving nonlinear Volterra integro-differential equations involving delay. We apply trapezium rule to the integral involved in the equation. Further, Daftardar-Gejji and Jafari method (DGJ) is employed to solve the implicit equation. Existence-uniqueness theorem is derived for solutions of such equations and the error and convergence analysis of the proposed method is presented. We illustrate efficacy of the newly proposed method by constructing examples.
In this paper we consider a class of fourth order nonlinear integro-differential equations with Navier boundary conditions. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.
In this paper, we consider a boundary value problem (BVP) for a fourth order nonlinear functional integro-differential equation. We establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for the total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.
In this work, a new technique has been presented to find approximate solution of linear integro-differential equations. The method is based on modified orthonormal Bernoulli polynomials and an operational matrix thereof. The method converts a given integro-differential equation into a set of algebraic equations with unknown coefficients, which is easily obtained with help of the known functions appearing in the equation, modified Bernoulli polynomials and operational matrix. Approximate solution is obtained in form of a polynomial of required degree. The method is also applied to three well kn…
High-dimensional parabolic partial integro-differential equations (PIDEs) appear in many applications in insurance and finance. Existing numerical methods suffer from the curse of dimensionality or provide solutions only for a given space-time point. This gave rise to a growing literature on deep learning based methods for solving partial differential equations; results for integro-differential equations on the other hand are scarce. In this paper we consider an extension of the deep splitting scheme due to Beck, Becker, Cheridito, Jentzen, Neufeld (2021) and Germain, Pham, Warin (2022) to PID…
In this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre. Based in the work by Hure, Pham and Warin, we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on Lèvy processes and a new neural network approximation of the nonlocal part to overcome the …
