scholarly journals ON THE UNIFORM CONVERGENCE OF THE APPROXIMATE SOLUTION OF SINGULAR INTEGRO-DIFFERENTIAL EQUATION OF THE FIRST KIND

2017 ◽  
Vol 19 (6) ◽  
pp. 54-60
Author(s):  
A.V. Ozhegova ◽  
L.E. Hayrullina
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Orthogonal Polynomials ◽  
Uniform Convergence ◽  
Boundary Problem ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Cauchy Kernel ◽  
Uniform Estimates ◽  
Stated Problem

The article presents the study of the boundary problem for singular inte-gro-differential equation of the first kind with Cauchy kernel. The authors introduce the pair of weight spaces to prove the correctness of the stated problem. The article states the sufficient conditions for the convergence of the general direct method, the method of orthogonal polynomials, and as a result uniform estimates for errors of approximate solution.

scholarly journals An approximate solution of one singular integro-differential equation using the method of orthogonal polynomials

2020 ◽  
pp. 86-96
Author(s):  
Galina A. Rasolko ◽  
Sergei M. Sheshko
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Orthogonal Polynomials ◽  
Electromagnetic Waves ◽  
Logarithmic Singularity ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Solution Function ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations

Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of H-polarized electromagnetic waves by a screen with a curved boundary, are constructed.  This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative.  The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

scholarly journals THE NUMERICAL SOLUTION BY THE METHOD OF DIRECT INTEGRALS OF DIFFERENTIATION OF EQUATIONS HAVE AN APPLICATION IN THE GAS FILTRATION THEOREM

2020 ◽  
pp. 147-153
Author(s):  
Abdujabbor Abdurazakov ◽  
Nasiba Makhmudova ◽  
Nilufar Mirzamakhmudova
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Error Estimate ◽  
Numerical Solution ◽  
Direct Method ◽  
Gas Filtration ◽  
Sweep Method ◽  
Time Step

On the basis of the direct method and a combination of differential sweep, the article developed a calculated algorithm for solving gas filtration, thereby taking into account the convergence of the approximate solution to the exact one. KEYWORDS: direct method, sweep method, differential equation, time step, convergence, approximate solution, error estimate.

A Procedure for Investigating the Liapunov Stability of Nonautonomous Linear Second-Order Systems

1973 ◽  
Vol 40 (4) ◽  
pp. 1103-1106 ◽  
Author(s):  
S. E. Jones ◽  
T. R. Robe
Keyword(s):  
Differential Equation ◽  
Ad Hoc ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Liapunov Function ◽  
Null Solution ◽  
Liapunov Stability ◽  
The Stability ◽  
Second Order Systems ◽  
Liapunov's Direct Method

Utilizing Liapunov’s direct method a procedure is presented which generates sufficient conditions for the stability of the null solution of the nonautonomous differential equation x¨ + b(t)x˙ + a(t)x = 0. This procedure systematically leads to the construction of a Liapunov function for a given differential equation and thus eliminates the normally ad hoc nature of the direct method. Four examples illustrating the procedure are discussed.

scholarly journals Uniform convergence of the collocation method for Prandtl's Integro-differential equation

2000 ◽  
Vol 42 (1) ◽  
pp. 151-168 ◽  
Author(s):  
M. R. Capobianco ◽  
G. Criscuolo ◽  
P. Junghanns ◽  
U. Luther
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Uniform Convergence ◽  
Collocation Method ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Quadrature Method ◽  
Integro Differential Equation ◽  
Spaces Of Continuous Functions

AbstractAn integro-differential equation of Prandtl's type and a collocation method as well as a collocation-quadrature method for its approximate solution is studied in weighted spaces of continuous functions.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

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.   

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals Oscillation and non-oscillation of some neutral differential equations of odd order

1992 ◽  
Vol 15 (3) ◽  
pp. 509-515 ◽  
Author(s):  
B. S. Lalli ◽  
B. G. Zhang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solution ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
Neutral Term ◽  
Odd Order ◽  
Existence Criterion ◽  
Oscillation Of Solutions

An existence criterion for nonoscillatory solution for an odd order neutral differential equation is provided. Some sufficient conditions are also given for the oscillation of solutions of somenth order equations with nonlinearity in the neutral term.

On Approximate Large Deviations for 1D Diffusion

2003 ◽  
Vol 10 (2) ◽  
pp. 381-399
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Approximation Scheme ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Euler Approximation ◽  
One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

scholarly journals A New Approach for Approximate Solution of ADE: Physical-Based Modeling of Carriers in Doping Region

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 458
Author(s):  
Leobardo Hernandez-Gonzalez ◽  
Jazmin Ramirez-Hernandez ◽  
Oswaldo Ulises Juarez-Sandoval ◽  
Miguel Angel Olivares-Robles ◽  
Ramon Blanco Sanchez ◽  
...  
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytic Solution ◽  
Order Differential Equation ◽  
Electric Circuit ◽  
New Approach ◽  
In Doping ◽  
Electric Behavior ◽  
Spatio Temporal ◽  
Physical Phenomena

The electric behavior in semiconductor devices is the result of the electric carriers’ injection and evacuation in the low doping region, N-. The carrier’s dynamic is determined by the ambipolar diffusion equation (ADE), which involves the main physical phenomena in the low doping region. The ADE does not have a direct analytic solution since it is a spatio-temporal second-order differential equation. The numerical solution is the most used, but is inadequate to be integrated into commercial electric circuit simulators. In this paper, an empiric approximation is proposed as the solution of the ADE. The proposed solution was validated using the final equations that were implemented in a simulator; the results were compared with the experimental results in each phase, obtaining a similarity in the current waveforms. Finally, an advantage of the proposed methodology is that the final expressions obtained can be easily implemented in commercial simulators.

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