On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups

2016 ◽  
Vol 56 (7) ◽  
pp. 1269-1292
Author(s):  
D. V. Gulua ◽  
J. L. Rogava
Keyword(s):  
Approximate Solution ◽  
Evolution Equation ◽  
Semidiscrete Scheme

Construction and numerical resolution of high-order accuracy decomposition scheme for a quasi-linear evolution equation

2018 ◽  
Vol 25 (3) ◽  
pp. 337-348
Author(s):  
Nana Dikhaminjia ◽  
Jemal Rogava ◽  
Mikheil Tsiklauri
Keyword(s):  
Approximate Solution ◽  
Evolution Equation ◽  
Continuous Operator ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
Lipschitz Continuous ◽  
Decomposition Scheme ◽  
Linear Evolution ◽  
Numerical Resolution ◽  
Linear Evolution Equation

AbstractIn the present work the Cauchy problem for an abstract evolution equation with a Lipschitz-continuous operator is considered, where the main operator represents the sum of positive definite self-adjoint operators. The fourth-order accuracy decomposition scheme is constructed for an approximate solution of the problem. The theorem on the error estimate of an approximate solution is proved. Numerical calculations for different model problems are carried out using the constructed scheme. The obtained numerical results confirm the theoretical conclusions.

scholarly journals ON APPROXIMATE SOLUTIONS OF SEMILINEAR EVOLUTION EQUATIONS

2004 ◽  
Vol 16 (03) ◽  
pp. 383-420 ◽  
Author(s):  
CARLO MOROSI ◽  
LIVIO PIZZOCCHERO
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Evolution Equation ◽  
Blow Up ◽  
Linear Part ◽  
Galerkin Approximation ◽  
Approximate Solutions ◽  
Nonlinear Heat Equation ◽  
Nonlinear Part ◽  
Control Equation

A general framework is presented to discuss the approximate solutions of an evolution equation in a Banach space, with a linear part generating a semigroup and a sufficiently smooth nonlinear part. A theorem is presented, allowing one to infer from an approximate solution the existence of an exact solution. According to this theorem, the interval of existence of the exact solution and the distance of the latter from the approximate solution can be evaluated by solving a one-dimensional "control" integral equation, where the unknown gives a bound on the previous distance as a function of time. For example, the control equation can be applied to the approximation methods based on the reduction of the evolution equation to finite-dimensional manifolds; among them, the Galerkin method is discussed in detail. To illustrate this framework, the nonlinear heat equation is considered. In this case the control equation is used to evaluate the error of the Galerkin approximation; depending on the initial datum, this approach either grants global existence of the solution or gives fairly accurate bounds on the blow up time.

Far-Field Behavior of Waves in Non-Ideal Magnetogasdynamics

2020 ◽  
Author(s):  
Ankita Sharma ◽  
Rajan Arora
Keyword(s):  
Approximate Solution ◽  
Evolution Equation ◽  
Asymptotic Analysis ◽  
Asymptotic Method ◽  
Nonlinear Waves ◽  
Numerical Technique ◽  
Ideal Gas ◽  
Far Field ◽  
Field Behavior ◽  
Hyperbolic Quasilinear System

We have presented a study on the far-field behavior of weak nonlinear waves in magnetogasdynamics. An asymptotic analysis is carried out for the study. An evolution equation is obtained by using an asymptotic method which helps in learning the far-field behavior of a hyperbolic quasilinear system governing the propagation of nonlinear waves in a non-ideal gas. A numerical technique MVIM is employed to obtain the approximate solution of the evolution equation.

Approximate Solution of One Dynamic Problem of Geoinformatics

2010 ◽  
Vol 42 (5) ◽  
pp. 1-11 ◽  
Author(s):  
Vladimir M. Bulavatskiy ◽  
Vasiliy V. Skopetsky
Keyword(s):  
Approximate Solution ◽  
Dynamic Problem

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

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