Methods and effective algorithms for solving multidimensional integral equations
https://doi.org/10.32362/2500-316X-2022-10-6-70-77
Abstract
Objectives. Integral equations have long been used in mathematical physics to demonstrate existence and uniqueness theorems for solving boundary value problems for differential equations. However, despite integral equations have a number of advantages in comparison with corresponding boundary value problems where boundary conditions are present in the kernels of equations, they are rarely used for obtaining numerical solutions of problems due to the presence of equations with dense matrices that arise that when discretizing integral equations, as opposed to sparse matrices in the case of differential equations. Recently, due to the development of computer technology and methods of computational mathematics, integral equations have been used for the numerical solution of specific problems. In the present work, two methods for numerical solution of two-dimensional and three-dimensional integral equations are proposed for describing several significant classes of problems in mathematical physics.
Methods. The method of collocation on non-uniform and uniform grids is used to discretize integral equations. To obtain a numerical solution of the resulting systems of linear algebraic equations (SLAEs), iterative methods are used. In the case of a uniform grid, an efficient method for multiplying the SLAE matrix by vector is created.
Results. Corresponding SLAEs describing the considered classes of problems are set up. Efficient solution algorithms using fast Fourier transforms are proposed for solving systems of equations obtained using a uniform grid.
Conclusions. While SLAEs using a non-uniform grid can be used to describe complex domain configurations, there are significant constraints on the dimensionality of described systems. When using a uniform grid, the dimensionality of SLAEs can be several orders of magnitude higher; however, in this case, it may be difficult to describe the complex configuration of the domain. Selection of the particular method depends on the specific problem and available computational resources. Thus, SLAEs on a non-uniform grid may be preferable for many two-dimensional problems, while systems on a uniform grid may be preferable for three-dimensional problems.
About the Author
A. B. Samokhin
MIREA – Russian Technological University
Russian Federation
Competing Interests:
Russian Federation
Alexander B. Samokhin - Dr. Sci. (Phys.-Math.), Professor, Applied Mathematics Department, Institute of Information Technology
78, Vernadskogo pr., Moscow, 119454
Scopus Author ID 7005200099, RSCI SPIN-code 6302-0596
Competing Interests:
The author declares no conflicts of interest
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While systems of linear algebraic equations (SLAEs) using a non-uniform grid can be used to describe complex domain configurations, there are significant constraints on the dimensionality of described systems. When using a uniform grid, the dimensionality of SLAEs can be several orders of magnitude higher; however, in this case, it may be difficult to describe the complex configuration of the domain. Selection of the particular method depends on the specific problem and available computational resources. Thus, SLAEs on a non-uniform grid may be preferable for many two-dimensional problems, while systems on a uniform grid may be preferable for three-dimensional problems.
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For citations:
Samokhin A.B. Methods and effective algorithms for solving multidimensional integral equations. Russian Technological Journal. 2022;10(6):70-77. https://doi.org/10.32362/2500-316X-2022-10-6-70-77
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