On the solution of one integro-differential equation with singular and hypersingular integrals

In this paper, we consider a new hypersingular integro-diﬀerential equation of arbitrary order on a closed curve located in the complex plane. The integrals in the equation are understood in the sense of the ﬁnite Hadamard part. The equation refers to linear integro-diﬀerential equations with variable coeﬃcients of a particular form. A characteristic feature of the equation is its representation with the help of determinants close to the Vronsky ones. The method of analytical continuation, properties of determinants, and generalized Sokhotsky formulas are used for the study. The equation reduces to the Riemann boundary value problem of a jump in a certain class of functions. If the Riemann boundary problem turns out to be solvable, then one should solve linear inhomogeneous diﬀerential equations in the class of analytic functions in the domains of the complex plane. The analysis of the obtained solutions in an inﬁnitely distant point is not evident. The study has a complete look. The conditions for the solvability of the original equation are explicitly written out. When they are fulﬁlled, the solution is explicitly written, to which an example is given.

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A hypersingular integro-differential equation with linear functions in coefficients

Шилин

2023

Vescì Akademìì navuk Belarusì. Seryâ fizika-matematyčnyh navuk

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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.

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On the solution of one integro-differential equation with singular and hypersingular integrals

Cited by 4 publications

References 5 publications

Solution of an Integrodifferential Equation Containing Linear Functions of a Special Form in the Coefficients

Solution of an Integrodifferential Equation Containing Linear Functions of a Special Form in the Coefficients

A solution of the hypersingular integro-diﬀerential equation with determinants of the Vronsky type

A hypersingular integro-differential equation with linear functions in coefficients

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