Parabolic Volterra Integro-differential Equation Research Articles

Error analysis and approximation of Jacobi pseudospectral method for the integer and fractional order integro-differential equation

Time-space Jacobi pseudospectral method is constructed to approximate the numerical solutions of the fractional Volterra integro-differential and parabolic Volterra integro-differential equations. We define fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi-Gauss-Lobatto points. The fractional derivative is defined in the modified Atangana-Baleanu derivative defined in the Caputo sense formula at JGL points. Further, we transform the domain of fractional Volterra integro-differential and parabolic Volterra integro-differential equations to the standard interval [−1,1] using variable transformation and function transformation. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The theory of error estimates and convergence analysis for the proposed method is also derived. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

A class of developed schemes for parabolic integro-differential equations

In this paper, we propose a class of methods to solve the parabolic Volterra integro-differential equations with bounded and unbounded domains. More precisely, we change the parabolic Volterra integro-differential equations to well-posed linear and nonlinear dynamical systems. Then, the obtained systems are solved by using a new class of algorithms consisting linear multi-step formulas in which these schemes are constructed through the hybrid of Gergory's formula, finite difference and multi-step methods. Error bounds are derived in both bounded and unbounded domains. Some numerical examples are then presented to illustrate the efficiency and accuracy of the proposed methods. Furthermore, stability and convergence of proposed methods are established and we denote the numerical simulations. Moreover, some tests are conducted on data with measurement noise to consider the performance of the proposed methods.

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.

Fully spectral collocation method for nonlinear parabolic partial integro-differential equations

The numerical approximation of solution to nonlinear parabolic Volterra and Fredholm partial integro-differential equations is studied in this paper. Unlike the conventional methods which discretize the time variable by finite difference schemes, we use the spectral method for this purpose. Indeed, both of the space and time discretizations are based on the Legendre-collocation method which lead to conversion of the problem to a nonlinear system of algebraic equations. The convergence of the proposed method is proven by providing an L∞ error estimate. Several numerical examples are included to demonstrate the efficiency and spectral accuracy of the proposed method in the space and time directions.

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.

The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

This article focuses on obtaining analytical solutions for d-dimensional, parabolic Volterra integro-differential equations with different types of frictional memory kernel. Based on Laplace transform and Fourier transform theories, the properties of the Fox-H function and convolution theorem, analytical solutions for the equations in the infinite domain are derived under three frictional memory kernel functions. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, the Fox-H function and the convolution form of the Fourier transform. In addition, graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equations with a power-law memory kernel. It can be seen that the solution curves are subject to Gaussian decay at any given moment.

Quasi Wavelet based numerical method for Volterra integro-differential equations on unbounded spatial domains

In this paper we study the numerical solution of the parabolic Volterra integro-differential equation with a memory term on one-dimensional unbounded domain. A new numerical method is proposed by applying the forward Euler scheme in time and the quasi-wavelet based numerical scheme in space. Some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.

Crank-Nicolson Quasi-Wavelet Based Numerical Method for Volterra Integro-Differential Equations on Unbounded Spatial Domains

AbstractThe numerical solution of a parabolic Volterra integro-differential equation with a memory term on a one-dimensional unbounded spatial domain is considered. A quasi-wavelet based numerical method is proposed to handle the spatial discretisation, the Crank-Nicolson scheme is used for the time discretisation, and second-order quadrature to approximate the integral term. Some numerical examples are presented to illustrate the efficiency and accuracy of this approach.

Controllability of a class of Volterra equations in Hilbert spaces with completely monotone kernel

We consider a parabolic Volterra integro-differential equation in Hilbert space with a completely monotone convolution kernel and a sectorial operator. Utilizing a state space setting, we show that for a large class of kernels the state cannot be controlled exactly to zero. On the other hand, equations of our type are always approximately controllable, provided the control operator has dense range.

The spectral methods for parabolic Volterra integro-differential equations

In this paper we study the numerical solutions to parabolic Volterra integro-differential equations in one-dimensional bounded and unbounded spatial domains. In a bounded domain, the given parabolic Volterra integro-differential equation is converted to two equivalent equations. Then, a Legendre-collocation method is used to solve them and finally a linear algebraic system is obtained. For an unbounded case, we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain. In both cases, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.

Artificial boundary conditions for parabolic Volterra integro-differential equations on unbounded two-dimensional domains

In this paper we study the numerical solution of parabolic Volterra integro-differential equations on certain unbounded two-dimensional spatial domains. The method is based on the introduction of a feasible artificial boundary and the derivation of corresponding artificial (fully transparent) boundary conditions. Two examples illustrate the application and numerical performance of the method.

The numerical solution of parabolic Volterra integro-differential equations on unbounded spatial domains

In this paper we study the numerical approximation of solutions to parabolic partial Volterra integro-differential equations in unbounded (one-dimensional) spatial domains. The key to these methods is the derivation of fully transparent artificial boundary conditions for some feasible, prescribed computational domain. Two numerical examples illustrate the application and the effectiveness of the artificial boundary method.

Runge-Kutta Time Discretizations of Parabolic Volterra Integro-Differential Equations

Runge-Kutta Time Discretizations of Parabolic Volterra Integro-Differential Equations

Parabolic Volterra Integrodifferential Equations of Convolution Type

Parabolic Volterra Integrodifferential Equations of Convolution Type

Weak solutions for a class of parabolic Volterra integrodifferential equations

We define weak solutions for a class of Volterra integrodifferential equations of the form u ′( t ) + Au ( t ) = ∝ 0 t a ( t , s ) g ( s , u ( s )) ds + f ( t , u ( t )), t ⩾ 0, u (0) = 0. The operator A is the negative infinitesimal generator of an analytic semigroup in a Banach space X . The operator g ( t , u ) is related to A by a special form g(t, u) = A 1 2 q(t, u) , where q ( t , u ) is an appropriate “lower order” operator. We show the existence and uniqueness of weak solutions and their continuability to infinity under suitable conditions. Using our results we study the asymptotic behavior, as time goes to infinity, of strong solutions of a second order initial-boundary value problem.

The Numerical Solution of a Class of Quasilinear Parabolic Volterra Equations Arising in Polymer Rheology

A class of quasilinear parabolic Volterra integrodifferential equations, which includes models for the stretching of filaments of molten polymers, is solved numerically. Due to the presence of small parameters and large relaxation constants, an implicit Euler type discretization scheme is chosen. A convergence proof for sufficiently small solutions is given, which also shows that the scheme preserves the asymptotic character of the solutions of the continuous problem as $t \pm \infty $. Computations are reported, which focus on a comparison of the solutions in cases where inertial, Newtonian and viscoelastic effects have various relative importance.