Existence and uniqueness of a solution to a partial integro-differential equation by the method of lines

Bahuguna, D.; Dabas, Jaydev

doi:10.14232/ejqtde.2008.1.4

Cited by 17 publications

(24 citation statements)

References 10 publications

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“…The classical papers are [7,8,22,23] where parabolic equations were considered. Existence results based on the method of lines can be found in [3,14,19,24,25]. Parabolic problems and first order partial differential equations and boundary-value problems for nonlinear elliptic equations were considered.…”

Section: Introductionmentioning

confidence: 99%

Method of Lines for Quasilinear Functional Differential Equations

Czernous

Kamont

2014

Ukr Math J

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We give a theorem on the error estimate of approximate solutions for ordinary functional differential equations. The error is estimated by a solution of an initial problem for nonlinear functional differential equation. We apply this general result to the investigation of the convergence of the numerical method of lines for evolution functional differential equations. Initial boundary-value problems for quasilinear equations are transformed by discretization in spatial variables into systems of ordinary functional differential equations. Nonlinear estimates of the Perron-type with respect to functional variables for given operators are assumed. Numerical examples are given. Наведено теорему про оцiнку похибки наближених розв'язкiв звичайних диференцiальних рiвнянь. Похибка оцiнюється за допомогою розв'язку початкової задачi для нелiнiйного функцiонально-диференцiального рiвняння. Цей загальний результат застосовується при дослiдженнi збiжностi числового методу лiнiй для еволюцiї функцiональнодиференцiальних рiвнянь. За допомогою дискретизацiї по просторових змiнних початково-крайовi задачi для квазiлiнiйних рiвнянь зводяться до систем звичайних диференцiальних рiвнянь. Припускається справедливiсть нелiнiйних оцiнок перронiвського типу вiдносно функцiональних змiнних для заданих операторiв. Наведено також чисельнi приклади.

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Section: Introductionmentioning

confidence: 99%

Method of Lines for Quasilinear Functional Differential Equations

Czernous

Kamont

2014

Ukr Math J

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“…The ordinary/partial differential equation along with the weighted integral of unknown function gives rise to an integro-differential equation (IDE) or a partial integro-differential equation (PIDE) respectively. Analysis of such equations can be found in [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Solving Partial Integro-Differential Equations Using Laplace Transform Method

Thorwe¹

2012

AJCAM

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Partialintegro-differential equations (PIDE) occur naturally in various fields of science, engineering and social sciences. In this article, we propose a most general form of a linear PIDE with a convolution kernel. We convert the proposed PIDE to an ordinary differential equation (ODE) using a Laplace transform (LT). Solving this ODE and applying inverse LT an exact solution of the problem is obtained. It is observed that the LT is a simple and reliable technique for solving such equations. A variety of numerical examples are presented to show the performance and accuracy of the proposed method.

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“…For instance, we refer to Rektorys [8] for a rich illustration of method applied to various interesting physical problems. Bahuguna and Dabas [1,2] have used the method of semidiscretization in time to study a partial differential equation with an integral condition involving delay arising in population dynamics.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal Parabolic Integro-differential Equations with Delays

kumar¹,

Kumar²,

Shukla³

2012

International Journal of Applied Mathematical Research

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In this paper we consider a parabolic integro-differential equation with delay and a nonlocal boundary condition. We apply the method of semidiscretization in time, also known as the method of lines, to establish the existence and uniqueness of the considered problem. We also establish the continuous dependence of the solution on the initial data. Finally, an application of the established results is demonstrated.

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Existence and uniqueness of a solution to a partial integro-differential equation by the method of lines

Cited by 17 publications

References 10 publications

Method of Lines for Quasilinear Functional Differential Equations

Method of Lines for Quasilinear Functional Differential Equations

Solving Partial Integro-Differential Equations Using Laplace Transform Method

Nonlocal Parabolic Integro-differential Equations with Delays

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