Quadrature linear integro-differential equations on a closed curve located in the complex plane are solved. The equations contain singular integrals which are understood in the sense of the main value and hypersingular integrals which are understood in the sense of the Hadamard finite part. The coefficients of the equations have a special structure.
In this paper, we investigated a new linear integro-differential equation of arbitrary order given on the closed curve located on a complex plane. The coefficients of the equation are variables and have a special form. The characteristic feature is the presence of linear functions in the coefficients. The equation is reduced to the consecutive solution of a Riemann boundary value problem on an initial curve and two linear differential equations. Differential equations are solved for analytic functions in areas into which the initial curve separates a complex plane. The corresponding fundamental systems of solutions are found, after that the arbitrary-constant variation method is applied. To achieve the analyticity of the obtained solutions the restrictions are imposed. All the arising conditions of resolvability of the input equation are written down explicitly, and if they are carried out then the solution is written in an explicit form. We represent the example demonstrating the existence of the cases when all conditions of resolvability are satisfied.
In this paper, we study an integro-differential equation on a closed curve located on the complex plane. The integrals included in the equation are understood as a finite part by Hadamard. The coefficients of the equation have a particular structure. The analytical continuation method is applied. The equation is reduced to a boundary value linear conjugation problem for analytic functions and linear Euler differential equations in the domains of the complex plane. Solutions of the Euler equations, which are unambiguous analytical functions, are sought. The conditions of solvability of the initial equation are given explicitly. The solution of the initial equation obtained under these conditions is also given explicitly. Examples are considered.
The linear hypersingular integro-differential equation of arbitrary order on a closed curve located on the complex plane is considered. A scheme is proposed to study this equation in the case when its coefficients have some particular structure. This scheme providers for the use of generalized Sokhotsky formulas, the solution of the Riemann boundary value problem and the solution in the class of analytical functions of linear differential equations. According to this scheme, the equations are explicitly solved, the coefficients of which contain power factors, so that along with the Riemann problem the arising differential equations are constructively solved. Solvability conditions, solution formulas, examples are given.
A new hypersingular integro-differential equation is considered on a closed curve located on the complex plane. The equation refers to linear equations with variable coefficients of a special kind. A characteristic feature is the presence of constant multipliers in the coefficients, given by some recurrent relations. The equation is first reduced to solving the Riemann boundary value problem on the original curve. A class of functions is established for solving the Riemann problem, after which this problem is solved. Next, it is necessary to solve two linear differential equations of arbitrary order for analytical functions in two different regions of the complex plane. The corresponding fundamental systems of solutions are found, after which the method of variation of arbitrary constants is used for the solution. Restrictions are imposed on the obtained solutions of differential equations in order to achieve their analyticity. As a result, all the resulting solvability conditions of the original equation are written explicitly. The solution of the original equation after solving the differential equations can be written explicitly. Solved the example.
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