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 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.

Periodic solutions of nonautonomous second-order differential equations with a p-Laplacian

Analysis ◽  
2017 ◽  
Vol 37 (1) ◽  
pp. 1-11
Author(s):  
Hairong Lian ◽  
Dongli Wang ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Second Order ◽  
Periodic Boundary Value Problem ◽  
Second Order Differential Equations

AbstractIn this paper, we study a periodic boundary value problem for a nonautonomous second-order differential equation with a

scholarly journals Fixed point theorems via Generalized WF-Contractions with applications

SeMA Journal ◽  
2021 ◽  
Author(s):  
Rosana Rodríguez-López ◽  
Rakesh Tiwari
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
New Class ◽  
Second Order Differential Equations ◽  
Fixed Point Result

AbstractThe aim of this paper is to introduce a new class of mixed contractions which allow to revise and generalize some results obtained in [6] by R. Gubran, W. M. Alfaqih and M. Imdad. We also provide an example corresponding to this class of mappings and show how the new fixed point result relates to the above-mentioned result in [6]. Further, we present an application to the solvability of a two-point boundary value problem for second order differential equations.

scholarly journals Second-order neutrosophic boundary-value problem

2021 ◽  
Author(s):  
Sandip Moi ◽  
Suvankar Biswas ◽  
Smita Pal(Sarkar)
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Differential Calculus ◽  
Boundary Value ◽  
Second Order ◽  
Numerical Examples ◽  
Different Types ◽  
First Time

AbstractIn this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation.

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