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.

scholarly journals The Laplace-Adomian-Pade Technique for the ENSO Model

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Yi Zeng
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
High Accuracy ◽  
Padé Method

The Laplace-Adomian-Pade method is used to find approximate solutions of differential equations with initial conditions. The oscillation model of the ENSO is an important nonlinear differential equation which is solved analytically in this study. Compared with the exact solution from other decomposition methods, the approximate solution shows the method’s high accuracy with symbolic computation.

Bilateral set-valued stochastic integral equations

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3253-3274
Author(s):  
Marek Malinowski ◽  
Donal O'Regan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Approximate Solutions ◽  
Existence And Uniqueness Theorem ◽  
Initial State ◽  
Stochastic Integral Equations ◽  
Stability Of The Solution

We investigate bilateral set-valued stochastic integral equations and these equations combine widening and narrrowing set-valued stochastic integral equations studied in literature. An existence and uniqueness theorem is established using approximate solutions. In addition stability of the solution with respect to small changes of the initial state and coefficients is established, also we provide a result on boundedness of the solution, and an estimate on a distance between the exact solution and the approximate solution is given. Finally some implications for deterministic set-valued integral equations are presented.

Time-of-flight diffraction method for joint with linear misalignment

2021 ◽  
Vol 63 (11) ◽  
pp. 654-658
Author(s):  
Y Kurokawa ◽  
T Kawaguchi ◽  
H Inoue
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Geometric Model ◽  
Time Of Flight ◽  
Diffraction Method ◽  
Approximate Solutions ◽  
Weld Joints ◽  
Time Of Flight Diffraction ◽  
Flaw Sizing ◽  
Probe Spacing

The time-of-flight diffraction (TOFD) method is known as one of the most accurate flaw sizing methods among the various ultrasonic testing techniques. However, the standard TOFD method cannot be applied to weld joints with linear misalignment because of its basic assumptions. In this study, a geometric model of the TOFD method for weld joints with linear misalignment is introduced and an exact solution for calculating the flaw tip depth is derived. Since the exact solution is extremely complex, a simple approximate solution is also derived assuming that the misalignment is sufficiently small relative to the probe spacing and the flaw tip depth. The error in the approximate solution is confirmed to be negligible if the assumptions are satisfied. Numerical simulations are conducted to assess the flaw sizing accuracy of both the exact and approximate solutions considering the constraint of the probe spacing and the influence of the excess metal shape. Finally, experiments are conducted to prove the applicability of the proposed method. As a result, the proposed method is proven to enable accurate flaw sizing of weld joints with linear misalignment.

PSEUDODIFFERENTIAL EQUATIONS ON THE SPHERE WITH SPHERICAL SPLINES

2011 ◽  
Vol 21 (09) ◽  
pp. 1933-1959 ◽  
Author(s):  
T. D. PHAM ◽  
T. TRAN ◽  
A. CHERNOV
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Theoretical Result ◽  
Galerkin Method ◽  
Unit Sphere ◽  
Approximate Solutions ◽  
Numerical Results ◽  
Pseudodifferential Equations ◽  
Optimal Convergence ◽  
Sobolev Norms

Spherical splines are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by using Galerkin method. We prove optimal convergence (in Sobolev norms) of the approximate solution by spherical splines to the exact solution. Our numerical results underlie the theoretical result.

scholarly journals Application of Jacobi Polynomials to the Approximate Solution of a Singular Integral Equation with Cauchy Kernel on the Real Half-line

2006 ◽  
Vol 6 (3) ◽  
pp. 326-335
Author(s):  
D. Pylak
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Jacobi Polynomials ◽  
Approximate Solutions ◽  
Half Line ◽  
Cauchy Kernel ◽  
The Real

AbstractIn this paper, exact solution of the characteristic equation with Cauchy kernel on the real half-line is presented. Next, Jacobi polynomials are used to derive approximate solutions of this equation. Moreover, estimations of errors of the approximated solutions are presented and proved.

scholarly journals Construction of a global solution for the one dimensional singularly-perturbed boundary value problem

2017 ◽  
Vol 8 (1-2) ◽  
pp. 52
Author(s):  
Samir Karasuljic ◽  
Enes Duvnjakovic ◽  
Vedad Pasic ◽  
Elvis Barakovic
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Uniform Convergence ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Singularly Perturbed ◽  
One Dimensional ◽  
The One ◽  
Theoretical Results

We consider an approximate solution for the one--dimensional semilinear singularly--perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an \(\varepsilon\)--uniform convergence of such gained the approximate solutions, in the maximum norm of the order \(\mathcal{O}\left(N^{-1}\right)\) on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has \(\varepsilon\)--uniform convergence, but now of order \(\mathcal{O}\left(\ln^2N/N^2\right)\) on \([0,1]\). In the end a numerical experiment is presented to confirm previously shown theoretical results.

scholarly journals A Galerkin-Petrov method for singular integral equations

1983 ◽  
Vol 25 (2) ◽  
pp. 261-275 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Convergence Theorem ◽  
Singular Integral Equations ◽  
Approximate Solutions ◽  
Principal Result ◽  
Cauchy Kernel

AbstractA Galerkin-Petrov method for the approximate solution of the complete singular integral equation with Cauchy kernel, based upon the use of two sets of orthogonal polynomials, is considered. The principal result of this paper proves convergence of the approximate solutions to the exact solution making use of a convergence theorem previously given by the author. In conclusion, some related topics such as a first iterate of the approximate solution and a discretized Galerkin-Petrov method are considered. The paper extends to a much more general equation many results obtained by other authors in particular cases.

A remark on Moore's new method of obtaining approximate solutions of the Dirac equation

1979 ◽  
Vol 57 (12) ◽  
pp. 2114-2119 ◽  
Author(s):  
Yasuo Tomishima
Keyword(s):  
Exact Solution ◽  
Wave Function ◽  
Approximate Solution ◽  
Dirac Equation ◽  
Approximate Solutions ◽  
Zero Order ◽  
New Method ◽  
Hydrogenic Atom ◽  
Approximate Wave Function ◽  
Approximate Wave

The new method for obtaining an approximate solution of the Dirac equation proposed by Moore is applied rigorously in zero order to the hydrogenic atom, and the results are compared with the exact solution and with those obtained by Pauli's approximation. It is concluded that up to the order of α2 the three methods give the same energy values, but the higher order contribution included in Moore's method makes the energy worse than Pauli's. Moore's zeroth approximate wave function is also examined.

scholarly journals A new approach for solving fractional RL circuit model through quadratic Legendre multi-wavelets

2018 ◽  
Vol 1 (1) ◽  
Author(s):  
Narottam Singh Chauhan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Error Analysis ◽  
Classical Solution ◽  
Convergence Theorem ◽  
Fractional Integration ◽  
Approximate Solutions ◽  
Mean Square ◽  
Legendre Wavelets ◽  
New Approach

The aim of present work is to obtain the approximate solution of fractional model for the electrical RL circuit by using quadratic Legendre multiwavelet method (QLMWM). The beauty of the paper is convergence theorem and mean square error analysis, which shows that our approximate solution converges very rapidly to the exact solution and the numerical solution is compared with the classical solution and Legendre wavelets method (LWM) solution, which is much closer to the exact solution. The fractional integration is described in the Riemann-Liouville sense. The results are shows that the method is very effective and simple. In addition, using plotting tools, we compare approximate solutions of each equation with its classical solution and LWM .

Symmetry Reduction and Analytical Solutions of Potential Kdv Equation

2017 ◽  
Vol 3 (05) ◽  
Author(s):  
Muhammet YÃœRÃœSOY
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Kdv Equation ◽  
Approximate Solutions ◽  
Series Solutions ◽  
Parameter Group ◽  
Classical Symmetries ◽  
The Kdv Equation ◽  
Different Types ◽  
Nonclassical Symmetries

In this paper, classical and nonclassical symmetries of the potential KdV equation are considered. A catalogue of symmetry reductions for potential KdV equation is obtained using the classical Lie method and nonclassical method due to Bluman and Cole.[1] The Lie algebra consists of five finite parameter Lie group transformations; two being the scaling symmetry and the others being translations. By using the nonclassical method, four finite parameter group transformations are obtained. Two different types of solutions, approximate and exact solutions, are found by using the symmetries. Using the classical symmetries, only approximate series solutions are constructed, but using the nonclassical symmetries, both the exact solution and the approximate solution were found. The advantage of nonclassical symmetries is that the exact solution of the KdV equation can be found with it. Perturbation method is also used for approximate solutions.

Sign in / Sign up

Export Citation Format

Share Document