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 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 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.

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 A hypersingular integro-differential equation of the Euler type

2020 ◽  
Vol 56 (1) ◽  
pp. 17-29 ◽  
Author(s):  
A. P. Shilin
Keyword(s):  
Differential Equation ◽  
Complex Plane ◽  
Continuation Method ◽  
Analytical Continuation ◽  
Closed Curve ◽  
Conjugation Problem ◽  
Linear Conjugation ◽  
Initial Equation ◽  
Integro Differential Equation ◽  
Analytical Functions

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.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

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.

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.

scholarly journals On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4013-4020
Author(s):  
Jianren Long ◽  
Sangui Zeng
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Complex Linear ◽  
Linear Differential ◽  
Growth Of Solutions

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

scholarly journals On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

scholarly journals Calculating Galois groups of third-order linear differential equations with parameters

2018 ◽  
Vol 20 (04) ◽  
pp. 1750038
Author(s):  
Andrei Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Galois Group ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Unipotent Radical ◽  
Third Order ◽  
Differential Galois Group ◽  
Bessel Differential Equation ◽  
Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

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