scholarly journals Solution of one hypersingular integro-differential equation defined by determinants

2021 ◽  
pp. 17-28
Author(s):  
Andrei P. Shilin
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Arbitrary Order ◽  
Exact Analytical Solution ◽  
Linear Equations ◽  
Linear Conjugation ◽  
Traditional Classification ◽  
Linear Differential ◽  
Problem Of Linear Conjugation

The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

scholarly journals ON THE CONSTRUCTION OF A FUNDAMENTAL SYSTEM OF SOLUTIONS OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS OF AN ARBITRARY ORDER

2020 ◽  
Vol 70 (2) ◽  
pp. 53-58
Author(s):  
P.B. Beisebay ◽  
◽  
G.H. Mukhamediev ◽  
Keyword(s):  
Differential Equation ◽  
Complex Analysis ◽  
Arbitrary Order ◽  
Fundamental System ◽  
Homogeneous Differential Equation ◽  
Complex Roots ◽  
Linear Homogeneous Differential Equation ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Homogeneous Linear

The paper proposes a method of presentation topics «On the construction of a fundamental system of solutions of a linear homogeneous differential equation with constant coefficients of an arbitrary order». In the traditional presentation of this topic in the case when the characteristic equation has complex roots, the particular solutions of the equation corresponding to them are constructed by applying the elements of complex analysis. In consequence of that, for students in the field, whose training programs included the theory of linear differential equations with constant coefficients and at the same time does not include the study of the theory of complex analysis, types of private solving the equation in this case is given without substantiation, or as a known fact, only for this case, previously issued elements complex analysis. Offered in the presentation technique differs from the traditional presentation of the topic in that it partial solutions scheme for constructing fundamental system of homogeneous linear equation with constant coefficients of arbitrary order is based only on the basis of the properties of the differential form corresponding to the left side of the equation, without using the elements of the theory of complex analysis.

scholarly journals Oscillator with a Sum of Noninteger-Order Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
L. Cveticanin ◽  
T. Pogány
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
Equations Of Motion ◽  
Dominant Term ◽  
Advantages And Disadvantages ◽  
Polynomial Type ◽  
Polynomial Nonlinearity ◽  
Non Linear ◽  
Linear Differential

Free and self-excited vibrations of conservative oscillators with polynomial nonlinearity are considered. Mathematical model of the system is a second-order differential equation with a nonlinearity of polynomial type, whose terms are of integer and/or noninteger order. For the case when only one nonlinear term exists, the exact analytical solution of the differential equation is determined as a cosine-Ateb function. Based on this solution, the asymptotic averaging procedure for solving the perturbed strong non-linear differential equation is developed. The method does not require the existence of the small parameter in the system. Special attention is given to the case when the dominant term is a linear one and to the case when it is of any non-linear order. Exact solutions of the averaged differential equations of motion are obtained. The obtained results are compared with “exact” numerical solutions and previously obtained analytical approximate ones. Advantages and disadvantages of the suggested procedure are discussed.

Asymptotic formulas for nonoscillatory solutions of conditionally oscillatory half-linear equations

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Zuzana Pátíková
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Order Differential Equation ◽  
Linear Equations ◽  
Continuous Functions ◽  
Second Order ◽  
Asymptotic Formulas ◽  
Nonoscillatory Solutions ◽  
Second Order Differential Equation ◽  
Linear Differential

AbstractWe establish asymptotic formulas for nonoscillatory solutions of a special conditionally oscillatory half-linear second order differential equation, which is seen as a perturbation of a general nonoscillatory half-linear differential equation $$ (r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = |x|^{p - 1} \operatorname{sgn} x,p > 1, $$ where r, c are continuous functions and r(t) > 0.

A Problem of Linear Conjugation for Analytic Functions with Boundary Values from the Zygmund Class

2002 ◽  
Vol 9 (2) ◽  
pp. 309-324
Author(s):  
V. Kokilashvili ◽  
V. Paatashvili
Keyword(s):  
Explicit Form ◽  
Analytic Functions ◽  
Boundary Values ◽  
Linear Conjugation ◽  
Zygmund Class ◽  
Solvability Conditions ◽  
Piecewise Continuous ◽  
Problem Of Linear Conjugation

Abstract The solvability conditions are established for a problem of linear conjugation for analytic functions with boundary values from the Zygmund class 𝐿(ln+ 𝐿) α when the conjugation coefficient is piecewise-continuous in the Hölder sense. Solutions of the problem are constructed in explicit form.

scholarly journals Value Distribution and Arbitrary-Order Derivatives of Meromorphic Solutions of Complex Linear Differential Equations in the Unit Disc

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 352
Author(s):  
Hai-Ying Chen ◽  
Xiu-Min Zheng
Keyword(s):  
Differential Equation ◽  
Unit Disc ◽  
Arbitrary Order ◽  
Value Distribution ◽  
Small Function ◽  
Meromorphic Solutions ◽  
Exponent Of Convergence ◽  
Complex Linear ◽  
Linear Differential ◽  
Derivatives Of

In this paper, we investigate the value distribution of meromorphic solutions and their arbitrary-order derivatives of the complex linear differential equation f ′ ′ + A ( z ) f ′ + B ( z ) f = F ( z ) in Δ with analytic or meromorphic coefficients of finite iterated p-order, and obtain some results on the estimates of the iterated exponent of convergence of meromorphic solutions and their arbitrary-order derivatives taking small function values.

scholarly journals On the Theory of Relaxation

1953 ◽  
Vol 1 (3) ◽  
pp. 101-110 ◽  
Author(s):  
A. R. Mitchell ◽  
D. E. Rutherford
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Numerical Method ◽  
Difference Equations ◽  
Linear Equations ◽  
Finite Difference Equations ◽  
Linear Differential

§ 1. When a numerical method of obtaining an approximate solution of a linear differential equation is employed, the process involves two distinct types of approximation. The region of integration having been covered with a regular net, the differential equation and the appropriate boundary conditions are replaced by finite difference equations which are linear equations in the values of the dependent variable at the nodes of the net.

On the asymptotic expansions of solutions of an nth order linear differential equation

1980 ◽  
Vol 85 (1-2) ◽  
pp. 15-57 ◽  
Author(s):  
R. B. Paris
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Asymptotic Expansions ◽  
Asymptotic Theory ◽  
Arbitrary Order ◽  
Order Linear Differential Equation ◽  
Linear Ordinary Differential Equations ◽  
Hypergeometric Type ◽  
Independent Variable ◽  
Linear Differential

SynopsisThe asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n are investigated for large values of the independent variable z in the complex plane. Solutions are expressed in terms of Mellin-Barnes integrals and their asymptotic expansions are subsequently determined by means of the asymptotic theory of integral functions of the hypergeometric type. Three classes of solutions are considered: (i) solutions whose behaviour is either exponentially large or algebraic for |z|→∞ in different sectors of the z-plane, (ii) solutions which are even and odd functions of z when the order n of the differential equation is even and (iii) solutions which are exponentially damped as |z|→∞ in a certain sector of the z-plane.

scholarly journals Adjoint differential equations

1927 ◽  
Vol 117 (776) ◽  
pp. 201-208
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Infinite Number ◽  
Adjoint Equation ◽  
Linear Equations ◽  
Point Of View ◽  
General Interest ◽  
Linear Difference Equations ◽  
Linear Differential

1. That adjoint differential equations have an analogue in the theory of linear difference equations seems to have been first observed by Bortolotti. The relation is essentially that of a matrix ǁ a rs ǁ to its transposed matrix ǁ a sr ǁ. It seems desirable, from this point of view, to carry out the transition from difference to differential equations, and thus prove that the analogy is a real one. This is done in Art. 2. There are further consequences of general interest. A set of linear equations corresponds to a differential equation and its boundary conditions, and thus we can find an interpretation of the adjoint boundary conditions introduced by Birkhoff into the theory of linear differential equations (Arts. 3-6). The relation between the two Green’s functions, implicit in Birkhoff’s work, then becomes evident (Art. 7). 2. We first prove that if the equations a r 1 y 1 + a r 2 y 2 + ... + a rn y n = fr ( r = 1 to n ) (1) are so constituted that they merge into the differential equation L ( y ) Ξ a m d m y / dx m . . . + a 1 dy / dx + a o y = f (2) by passing to an infinite number of infinitesimally spaced unknowns, the transposed equations a 1 r z 1 + a 2 r z 2 + ... + a nr z n = g r (3) merge into the adjoint equation M ( z ) Ξ (—) m d m / dx m ( a m z ) + ... - d / dx ( a 1 z ) + a o z = g .(4)

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