A Numerical Solution of Parabolic-Type Volterra Partial Integro-differential Equations by Laguerre Collocation Method

Abstract

Partial integro-differential equations occur in many fields of science and engineering. Besides, the class of parabolic-type differential equations is modelled in compression of poro-viscoelastic media, reaction-diffusion problems and nuclear reactor dynamics. In recent years, most mathematical models used in many problems of physics, biology, chemistry and engineering are based on integral and integro-differential equations. In this work, we propose a new effective numerical scheme based on the Laguerre matrix-collocation method to obtain the approximate solution of one dimensional parabolic-type Volterra partial integro-differential equations with the initial and boundary conditions. The presented method reduces the solution of the mentioned partial integro-differential equation to the solution of a matrix equation corresponding to system of algebraic equations with unknown Laguerre coefficients. Also, some numerical examples together with error estimation are presented to illustrate the validity and applicability of the proposed scheme.

Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials

“…Laguerre polynomials are used to solve some integer order integro-differential equations. These equations are given as Altarelli-Parisi equation (Kobayashi et al 1995), Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equation (Schoeffel 1999), Pantograph-type Volterra integrodifferential equation (Yüzbaşı 2014), linear Fredholm integro-differential equation (Baykus Savasaneril & Sezer 2016;Gürbüz et al 2014), linear integro-differential equation (Al-Zubaidy 2013), parabolic-type Volterra partial integro-differential equation (Gürbüz & Sezer 2017a), nonlinear partial integro-differential equation (Gürbüz & Sezer 2017b), delay partial functional differential equation (Gürbüz & Sezer 2017c). Besides, Laguerre polynomials are used to solve the fractional integro-differential equation (Mahdy & Shwayyea 2016).…”

Adaptive Finite element solution for Volterra partial integro differential equations

“…This type of parabolic PIDEs is utilized in modeling several phenomena in applied science [1,2,3]. Various methods have been utilized to solve parabolic PIDEs [4,5,6,7,8,9,10,11,12,13]. One of these methods is the finite element method (FEM) which is considered an effective method employed for obtaining approximate solutions to different types of problems arising in engineering applications [14,15,16,17].…”

“…12) estimates ∥e h ∥ l 2 . This estimator was proposed by Wiberget[20] and called element patch recovery.…”

A method for fractional Volterra integro-differential equations by Laguerre polynomials

“…Laguerre polynomials are used to solve some integer order integro-differential equations. These equations are given as Altarelli-Parisi equation [42], Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equation [43], pantograph-type Volterra integro-differential equation [44], linear Fredholm integro-differential equation [45,46], linear integrodifferential equation [47], parabolic-type Volterra partial integro-differential equation [48], nonlinear partial integro-differential equation [49], delay partial functional differential equation [50]. Besides, Laguerre polynomials are used to solve the fractional Fredholm integro-differential equation [51].…”