scholarly journals Numerical Solution for the 2D Linear Fredholm Functional Integral Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Neda Khaksari ◽  
Mahmoud Paripour ◽  
Nasrin Karamikabir
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Unknown Function ◽  
Numerical Solutions ◽  
Functional Integral ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

In this work, a numerical method is applied for obtaining numerical solutions of Fredholm two-dimensional functional linear integral equations based on the radial basis function (RBF). To find the approximate solutions of these types of equations, first, we approximate the unknown function as a finite series in terms of basic functions. Then, by using the proposed method, we give a formula for determining the unknown function. Using this formula, we obtain a numerical method for solving Fredholm two-dimensional functional linear integral equations. Using the proposed method, we get a system of linear algebraic equations which are solved by an iteration method. In the end, the accuracy and applicability of the proposed method are shown through some numerical applications.

scholarly journals On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 234
Author(s):  
Vladimir Vasilyev ◽  
Nikolai Eberlein
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Mellin Transform ◽  
Algebraic Equations ◽  
Conjugation Problem ◽  
Solvability Conditions ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

scholarly journals NUMERICAL CALCULATION OF NONSTATIONARY FRACTIONAL DIFFERENTIAL EQUATION IN PROBLEMS OF MODELING TOXIC SUBSTANCES DISTRIBUTION IN GROUND WATERS

2019 ◽  
pp. 70-80
Author(s):  
Alexandra Alekseyevna Afanasyeva ◽  
Tatyana Nikolayevna Shvetsova-Shilovskaya ◽  
Dmitriy Evgenevich Ivanov ◽  
Denis Igorevich Nazarenko ◽  
Elena Victorovna Kazarezova
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Differential Equations ◽  
Difference Scheme ◽  
Numerical Method ◽  
Fractional Differential Equation ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

scholarly journals An analytical model for simulating two-dimensional multispecies plume migration

2016 ◽  
Vol 20 (2) ◽  
pp. 733-753 ◽  
Author(s):  
Jui-Sheng Chen ◽  
Ching-Ping Liang ◽  
Chen-Wuing Liu ◽  
Loretta Y. Li
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Equation System ◽  
Screening Tools ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Fourier Cosine ◽  
Advection Dispersion ◽  
Linear Algebraic Equations

Abstract. The two-dimensional advection-dispersion equations coupled with sequential first-order decay reactions involving arbitrary number of species in groundwater system is considered to predict the two-dimensional plume behavior of decaying contaminant such as radionuclide and dissolved chlorinated solvent. Generalized analytical solutions in compact format are derived through the sequential application of the Laplace, finite Fourier cosine, and generalized integral transform to reduce the coupled partial differential equation system to a set of linear algebraic equations. The system of algebraic equations is next solved for each species in the transformed domain, and the solutions in the original domain are then obtained through consecutive integral transform inversions. Explicit form solutions for a special case are derived using the generalized analytical solutions and are compared with the numerical solutions. The analytical results indicate that the analytical solutions are robust, accurate and useful for simulation or screening tools to assess plume behaviors of decaying contaminants.

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