This article presents a computational method for solving a problem with parameter for a system of Fredholm integro-differential equations. Some additional parameters are introduced and the problem under consideration ...
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This article presents a computational method for solving a problem with parameter for a system of Fredholm integro-differential equations. Some additional parameters are introduced and the problem under consideration is reduced to solving a system of linear algebraic equations. The coefficients and right-hand side of the system are calculated by solving the Cauchy problems for ordinary differential equations. We establish a criterion for the unique solvability of the problem under consideration. A numerical algorithm is offered for solving the problem with parameter. The results are illustrated by numerical examples.
The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-diff...
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The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.
The paper devoted to the boundary value problem with parameter for second order system of hyperbolic equations. We study of a questions for existence and uniqueness of solution to the problem and a construction of alg...
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The paper devoted to the boundary value problem with parameter for second order system of hyperbolic equations. We study of a questions for existence and uniqueness of solution to the problem and a construction of algorithms for finding its solution. Conditions for the unique solvability to problem with parameter are established in the terms of fundamental matrix and initial data.
A linear boundary value problem with a parameter for a system of essentially loaded differential equations is investigated by the D. S. Dzhumabaev's parametrization method. A numerical algorithm is offered for sol...
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A linear boundary value problem with a parameter for a system of essentially loaded differential equations is investigated by the D. S. Dzhumabaev's parametrization method. A numerical algorithm is offered for solving the problem under consideration. The results are illustrated by numerical examples.
A problem of solvability with parameter for a differential-algebraic equation is considered. For solving the problem is applied Weierstrass canonical form. problem is reduced to an initial value problem with parameter...
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A problem of solvability with parameter for a differential-algebraic equation is considered. For solving the problem is applied Weierstrass canonical form. problem is reduced to an initial value problem with parameter for differential equations. Conditions for the existence and uniqueness of the problem with parameter for differential-algebraic equations are established.
We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearit...
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We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let (u) over tilde (x) be a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting v(x) = u(x) - (u) over tilde (x), we reduce the original problem to a problem with homogeneous boundary condition. The spectrum of the transformed problem consists of the values of the parameter for which this problem has a non-zero solution (the function v(x) = 0 is a solution for all values of the parameter). Under certain additional restrictions we construct an iterative process converging to a minimal semiregular solution of the transformed problem for an appropriately chosen starting point. We prove that any non-empty spectrum of the boundary-value problem is a ray [lambda*, +infinity), where lambda* > 0. As an application, we consider the Gol'dshtik mathematical model for separated flows of an incompressible fluid. We show that it satisfies the hypotheses of our theorem and has a non-empty spectrum.
We consider the nonlocal problem for fourth-order loaded hyperbolic equations with two independent variables. This problem is reduced to an equivalent problem consisting of a nonlocal problem for a system of loaded hy...
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We consider the nonlocal problem for fourth-order loaded hyperbolic equations with two independent variables. This problem is reduced to an equivalent problem consisting of a nonlocal problem for a system of loaded hyperbolic equations of the second order with functional parameters and integral relations by the method of introducing new unknown functions. Algorithms for finding solution to the equivalent problem are proposed. Conditions for well-posedness to the nonlocal problem for the system of loaded hyperbolic equations of the second order are obtained. Conditions for the existence of a unique classical solution to the nonlocal problem for fourth-order loaded hyperbolic equations are established.
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