scholarly journals Hypersingular integro-differential equations with power factors in coefficients

2019 ◽  
pp. 48-56 ◽  
Author(s):  
Andrei P. Shilin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Arbitrary Order ◽  
Closed Curve ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Solvability Conditions ◽  
Analytical Functions ◽  
Linear Differential

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.

scholarly journals Riemann’s differential boundary-value problem and its application to integro-differential equations

2019 ◽  
Vol 63 (4) ◽  
pp. 391-397 ◽  
Author(s):  
Andrei P. Shilin
Keyword(s):  
Boundary Condition ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Closed Curve ◽  
Riemann Boundary ◽  
Analytical Functions ◽  
Linear Differential ◽  
Derivatives Of

The boundary-value problem for analytical functions is investigated. The boundary condition is placed on a closed curve located on the complex plane. The problem belongs to the type of the generalized Riemann boundary-value problems. The boundary condition contains derivatives of the required functions. The problem is reduced to the usual Riemann problem and linear differential equations. The solution is built in closed form. The application of the solved problem to integro-differential equations is indicated.

scholarly journals Explicit solution of one hypersingular integro-differential equation of the second order

2019 ◽  
pp. 67-72 ◽  
Author(s):  
Andrei P. Shilin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Second Order ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Hypersingular Integrals ◽  
Linear Differential ◽  
Derivatives Of

The linear equation on the curve located on the complex plane is studied. The equation contains the desired function, its derivatives of the first and second orders, as well as hypersingular integrals with the desired function. The coefficients of the equation have a special structure. The equation is reduced to the Riemann boundary value problem for analytic functions and two second order linear differential equations. The boundary value problem is solved by Gakhov formulas, and the differential equations are solved by the method of variation of arbitrary constants. The solution of the original equation is constructed in quadratures. The result is formulated as a theorem. An example is given.

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

2020 ◽  
Vol 56 (3) ◽  
pp. 298-309
Author(s):  
A. P. Shilin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Complex Plane ◽  
Continuation Method ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Hypersingular Integrals ◽  
Integro Differential Equation ◽  
Linear Differential ◽  
Special Points

A linear integro-differential equation of the first order given on a closed curve located on the complex plane is studied. The coefficients of the equation have a special structure. The equation contains a singular integral, which can be understood as the main value by Cauchy, and a hypersingular integral which can be understood as the end part by Hadamard. The analytical continuation method is applied. The equation is reduced to a sequential solution of the Riemann boundary value problem and two linear differential equations. The Riemann problem is solved in the class of analytic functions with special points. Differential equations are solved in the class of analytical functions on the complex plane. The conditions for the solvability of the original equation are explicitly given. The solution of the equation when these conditions are fulfilled is also given explicitly. Examples are considered. A non-obvious special case is analyzed.

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

2021 ◽  
Vol 57 (3) ◽  
pp. 236-310
Author(s):  
A. P. Shilin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Complex Plane ◽  
Arbitrary Order ◽  
Variable Coefficients ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Integro Differential Equation ◽  
Riemann Boundary Problem ◽  
Class Of Functions

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.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

The use of Riemann problems in solving a class of transcendental equations

1973 ◽  
Vol 73 (1) ◽  
pp. 111-118 ◽  
Author(s):  
E. E. Burniston ◽  
C. E. Siewert
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Explicit Solutions ◽  
Riemann Boundary Value Problem ◽  
Riemann Problems ◽  
Riemann Boundary ◽  
Canonical Solution ◽  
Transcendental Equations ◽  
Explicit Expressions

AbstractA method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

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