On a problem for an elliptic type equation of the second kind with a conormal and integral condition

When approximating multidimensional partial differential equations, the values of the grid functions from neighboring layers are taken from the previous time layer or approximation. As a result, along with the approximation discrepancy, an additional discrepancy of the numerical solution is formed. To reduce this discrepancy when solving a stationary elliptic equation, parabolization is carried out, and the resulting equation is solved by the method of successive approximations. This discrepancy is eliminated in the approximate analytical method proposed below for solving two-dimensional equations of parabolic and elliptic types, and an exact solution of the system of finite difference equations for a fixed time is obtained. To solve problems with a boundary condition of the first kind on the first coordinate and arbitrary combinations of the first, second and third kinds of boundary conditions on the second coordinate, it is proposed to use the method of straight lines on the first coordinate and ordinary sweep method on the second coordinate. Approximating the equations on the first coordinate, a matrix equation is built relative to the grid functions. Using eigenvalues and vectors of the three-diagonal transition matrix, linear combinations of grid functions are compiled, where the coefficients are the elements of the eigenvectors of the three-diagonal transition matrix. Boundary conditions, and for a parabolic equation, initial conditions are formed for a given combination of grid functions. The resulting one-dimensional differential-difference equations are solved by the ordinary sweep method. From the resulting solution, proceed to the initial grid functions. The method provides a second order of approximation accuracy on coordinates. And the approximation accuracy in time when solving the parabolic equation can be increased to the second order using the central difference in time. The method is used to solve heat transfer problems when the boundary conditions are expressed by smooth and discontinuous functions of a stationary and non-stationary nature, and the right-hand side of the equation represents a moving source or outflow of heat.

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“…A similar series of calculations were carried out using an algorithm and a program to solve a parabolic equation [20].…”

Section: Computational Experimentsmentioning

confidence: 99%

Exact method to solve of linear heat transfer problems

2021

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“…Many problems in modern sciences, technology and economics are described by differential equations, the solutions of which are functions with first kind discontinuities at fixed or non-fixed times. Such differential equations are called differential equations with impulse effects [1][2][3][4][5][6][7][8]. As is known, in recent years the interest in the study of differential equations with nonlocal boundary conditions has increased (see, for example, [9][10][11][12][13][14][15][16][17]).…”

Section: Introductionmentioning

confidence: 99%

On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima

Yuldashev¹,

Fayziev²

2022

Nanosystems: Phys. Chem. Math.

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A nonlocal boundary value problem for a system of ordinary integro-differential equations with impulsive effects, degenerate kernel and maxima is investigated. The boundary value problem is given by the integral condition. The method of successive approximations in combination with the method of compressing mapping is used. The existence and uniqueness of the solution of the boundary value problem are proved. The continuous dependence of the solution on the right-hand side of the boundary value condition is shown. KEYWORDS impulsive integro-differential equations, nonlocal condition, successive approximations, existence and uniqueness, continuous dependence of solution FOR CITATION Yuldashev T.K., Fayziev A.K. On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima.

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“…x Проведенный анализ состояния дел в этом направлении показывает, что несмотря на то, что в настоящее время известны фундаментальные решения для более общих сингулярных уравнений, краевые задачи с условием Пуанкаре -Трикоми для вырождающихся уравнений эллиптического и эллиптико-гиперболического типов второго рода изучены сравнительно мало. Отметим лишь работы [18,19]. Монография [20] посвящена постановке и исследованию локальных и нелокальных краевых задач, в том числе, задаче Пуанкаре -Трикоми для уравнений смешанного типа с двумя линиями и различными порядками вырождения.…”

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Poincare–tricomi Problem for the Equation of a Mixed Elliptico-Hyperbolic Type of Second Kind

Abdullayev¹,

Ergashev²

2020

Tomsk State University Journal of Mathematics and Mechanics

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On a problem for an elliptic type equation of the second kind with a conormal and integral condition

Cited by 8 publications

References 3 publications

Exact method to solve of linear heat transfer problems

Exact method to solve of linear heat transfer problems

On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima

Poincare–tricomi Problem for the Equation of a Mixed Elliptico-Hyperbolic Type of Second Kind

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