Solution Of Partial Integro-differential Equation Research Articles

Numerical solution of partial integro-differential equation with a weakly singular kernel based on Sinc methods

We study the formulation of a space–time Sinc collocation method based on single and double exponential transformations for the numerical solution of the partial integro-differential equation with a weakly singular kernel in one or two space dimensions. The space–time adjective means that the Sinc collocation technique is employed simultaneously in time and space. The error bounds are given which show the exponential convergence rate of the method. Numerical results reported show that the validity and effectiveness of the proposed algorithms are attained for singular problems.

On the Approximate Solution of Partial Integro-Differential Equations Using the Pseudospectral Method Based on Chebyshev Cardinal Functions

In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an efficient method to solve PIDEs, especially for the integral part. Unlike when using Chebyshev functions, when using Chebyshev cardinal functions it is no longer necessary to integrate to find expansion coefficients of a given function. This reduces the computation. The convergence analysis is investigated and some numerical examples guarantee our theoretical results. We compare the presented method with others. The results confirm the efficiency and accuracy of the method.

Convergence of numerical schemes for the solution of partial integro-differential equations used in heat transfer

Integro-differential equations play an important role in may physical phenomena. For instance, it appears in fields like fluid dynamics, biological models and chemical kinetics. One of the most important physical applications is the heat transfer in heterogeneous materials, where physician are looking for efficient methods to solve their modeled equations. The difficulty of solving integro-differential equations analytically made mathematician to search about efficient methods to find an approximate solution. The present article is designed to supply numerical solution of a parabolic Volterra integro-differential equation under initial and boundary conditions. We have made an attempt to develop a numerical solution via the use of Sinc-Galerkin method, the convergence analysis via the use of fixed point theory has been discussed, and showed to be of exponential order. For comparison purposes, we approximate the solution of integro-differential equation using Adomian decomposition method. Sometimes, the Adomian decomposition method is a highly efficient technique used to approximate analytical solution of differential equations, applicability of Adomian decomposition method to partial integro-differential equations has not been studied in details previously in the literatures. In addition, we present numerical examples and comparisons to support the validity of these proposed methods.

Solution of Partial Integro-Differential Equations Involving Mixed Partial Derivatives by Laplace Substitution Method

This paper studies the Laplace substitution method for nonlinear partial integro-differential equations involving mixed partial derivatives and further applications of same method for coupled nonlinear partial integro-differential equations involving mixed partial derivatives.

ADI schemes for valuing European options under the Bates model

This paper is concerned with the adaptation of alternating direction implicit (ADI) time discretization schemes for the numerical solution of partial integro-differential equations (PIDEs) with application to the Bates model in finance. Three different adaptations are formulated and their (von Neumann) stability is analyzed. Ample numerical experiments are provided for the Bates PIDE, illustrating the actual stability and convergence behaviour of the three adaptations.

Fourth-Order Compact Finite Difference Scheme for American Option Pricing Under Regime-Switching Jump-Diffusion Models

In this paper, a fourth-order compact finite difference scheme is proposed for the solution of partial integro-differential equation arising in option pricing under jump-diffusion models. In proposed compact scheme, second derivative approximations of the unknowns are eliminated with the unknowns itself and their first derivative approximations while retaining the fourth order accuracy and tri-diagonal nature of the scheme. We use the proposed compact scheme with three time levels along with operator splitting technique to solve the linear complementary problem which arises in American option pricing under regime-switching jump-diffusion models. Moreover, it is shown that the proposed scheme leads to a tri-diagonal system of linear equations and fourth order accuracy is obtained. Since initial condition is not smooth enough for jump-diffusion model, we employ the smoothing operators to ensure high-order convergence rate. Numerical examples for American option pricing under Merton and Kou regime-switching jump-diffusion models are given.

Numerical Solution of Partial Integro-Differential Equations by using Projection Method

In this paper we study the numerical solution of initial-boundary problem for parabolic Volterra integro-differential equations in one dimensional. These equations include the partial differentiation of an unknown function and the integral term containing the unknown function which is the memory of problem. We have made an attempt to develop a method for Wavelet Galerkin which provides the approximate solution. Some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.

On the solutions of partial integrodifferential equations of fractional order

The main purpose of this paper is to study the existence of solutions for the nonlinear fractional partial integrodifferential equations with Dirichlet boundary condition. Under suitable assumption the results are established by using the Leray-Schauder fixed point theorem and Arzela-Ascoli theorem. An example is provided to illustrate the main result.

Operational matrix approach for the solution of partial integro-differential equation

In this paper, an effective numerical method is introduced for the treatment of Volterra singular partial integro-differential equations. They are based on the operational and almost operational matrix of integration and differentiation of 2D shifted Legendre polynomials. The methods convert the singular partial integro-differential equation in to a system of algebraic equations. Convergence analysis and error estimates are derived for the proposed method. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Pricing approximations and error estimates for local Lévy-type models with default

We find approximate solutions of partial integro-differential equations, which arise in financial models when defaultable assets are described by general scalar Lévy-type stochastic processes. We derive rigorous error bounds for the approximate solutions. We also provide numerical examples illustrating the usefulness and versatility of our methods in a variety of financial settings.

Solution of partial integro-differential equations by Elzaki transform method

Solution of partial integro-differential equations by Elzaki transform method

Unconditional Positive Stable Numerical Solution of Partial Integrodifferential Option Pricing Problems

This paper is concerned with the numerical solution of partial integrodifferential equation for option pricing models under a tempered stable process known as CGMY model. A double discretization finite difference scheme is used for the treatment of the unbounded nonlocal integral term. We also introduce in the scheme the Patankar-trick to guarantee unconditional nonnegative numerical solutions. Integration formula of open type is used in order to improve the accuracy of the approximation of the integral part. Stability and consistency are also studied. Illustrative examples are included.

SMOOTH SOLUTION OF PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS USING RADIAL BASIS FUNCTIONS

In this work, we present the method based on radial basis functions to solve partial integro-differential equations. We focus on the parabolic type of integro-differential equations as the most common forms including the ``\emph{memory}'' of the systems. We propose to apply the collocation scheme using radial basis functions to approximate the solutions of partial integro-differential equations. Due to the presented technique, system of linear or nonlinear equations is made instead of primary problem. The method is efficient because the rate of convergence of collocation method based on radial basis functions is exponential. Some numerical examples and investigation of the experimental results show the applicability and accuracy of the method.

On The Numerical Solution of Partial integro-differential equations

In this paper, we consider the approximate solution of the partial integro-differential equation. To solve this problem, we introduce a new nonstandard time discretization scheme. Then the fourth order finite difference and collocation method is presented for the numerical solution of this type of partial integro-differential equation (PIDE). A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed-form difference scheme. The efficiency and accuracy of the scheme is validated by its application to one test problem which have exact solutions. Numerical results show that this fourth-order scheme has the expected accuracy. The most advantages of compact finite difference method for PIDE are that it obtains high order of accuracy, while the time complexity to solve the matrix equations after we use compact finite difference method on PIDE is O(N), and it can solve very general case of PIDE.

Double Discretization Difference Schemes for Partial Integrodifferential Option Pricing Jump Diffusion Models

A new discretization strategy is introduced for the numerical solution of partial integrodifferential equations appearing in option pricing jump diffusion models. In order to consider the unknown behaviour of the solution in the unbounded part of the spatial domain, a double discretization is proposed. Stability, consistency, and positivity of the resulting explicit scheme are analyzed. Advantages of the method are illustrated with several examples.

A Reduced Basis for Option Pricing

We introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations (PIDEs) which arise in option pricing theory. Our method constructs the solution as a linear combination of basis functions constructed from a sequence of Black–Scholes solutions with different volatilities. We show that this a priori choice of basis leads to a sparse representation of option pricing functions, yielding an approximation error which decays exponentially in the number of basis functions. A Galerkin method using this basis for solving the pricing PDE is shown to have better numerical performance relative to commonly used finite-difference and finite-element methods for the CEV diffusion model and the Merton jump diffusion model. We also compare our method with a numerical proper orthogonal decomposition (POD). Finally, we show that this approach may be used advantageously for the calibration of local volatility functions.

Efficient solution of a class of partial integro-differential equation in reproducing kernel space

In [Y.l. Wang, T. Chaolu, Z. Chen, Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems, Int. J. Comput. Math. 87(2) (2010), pp. 367–380], we present three algorithms to solve a class of ordinary differential equations boundary value problems in reproducing kernel space. It is worth noting that our methods can get the solution of partial integro-differential equation. In this note, we use method 2 [M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83(1) (2006), pp. 123–129] to solve a class of partial integro-differential equation in reproducing kernel space. Numerical example shows our method is effective and has high accuracy.

Numerical Solutions of PIDEs for the Prices of Bond Options, Swaps, Caps and Floors for Levy-Based Stochastic Interest Rate Models

In this paper, we show numerically how to calculate the price of bond options, swaps, caps and floors for Levy one-factor stochastic interest rate models via partial integro-differential equations (PIDE). These models include, in particular, Ornshtein-Uhlenbeck (1930), Vasicek (1977), Cox-Ingersoll-Ross (1985), continuous-time GARCH, Ho-Lee (1986) and Hull-White (1990) models. We use Matache et al (2006) approach to solve numerically the PIDEs.

A Reduced Basis for Option Pricing

We introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations which arise in option pricing theory. Our method uses a basis of functions constructed from a sequence of Black-Scholes solutions with different volatilities. We show that this choice of basis leads to a sparse representation of option pricing functions, yielding an approximation whose precision is exponential in the number of basis functions. A Galerkin method using this basis for solving the pricing PDE is presented. Numerical tests based on the CEV diffusion model and the Merton jump diffusion model show that the method has better numerical performance relative to commonly used finite-difference and finite-element methods. We also compare our method with a numerical Proper Orthogonal Decomposition (POD). Finally, we show that this approach may be used advantageously for the calibration of local volatility functions.

Asymptotic Self-Similarity for Solutions of Partial Integro-Differential Equations

The question is studied whether weak solutions of linear partial integro-differential equations approach a constant spatial profile after rescaling, as time goes to infinity. The possible limits and corresponding scaling functions are identified and are shown to actually occur. The limiting equations are fractional diffusion equations which are known to have self-similar fundamental solutions. For an important special case, is is shown that the asymptotic profile is Gaussian and convergence holds in $L^2$, that is, solutions behave like fundamental solutions of the heat equation to leading order. Systems of integro-differential equations occurring in viscoelasticity are also discussed, and their solutions are shown to behave like fundamental solutions of a related Stokes system. The main assumption is that the integral kernel in the equation is regularly varying in the sense of Karamata.