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 .

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.

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 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 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 Error Analysis of Galerkin's Method for Semilinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Tadashi Kawanago
Keyword(s):  
Partial Differential Equations ◽  
Error Analysis ◽  
Convergence Theorem ◽  
Existence Result ◽  
Approximate Solutions ◽  
Numerical Verification ◽  
Semilinear Equations ◽  
Verification Methods ◽  
General Existence

We establish a general existence result for Galerkin's approximate solutions of abstract semilinear equations and conduct an error analysis. Our results may be regarded as some extension of a precedent work (Schultz 1969). The derivation of our results is, however, different from the discussion in his paper and is essentially based on the convergence theorem of Newton’s method and some techniques for deriving it. Some of our results may be applicable for investigating the quality of numerical verification methods for solutions of ordinary and partial differential equations.

scholarly journals Sampling-based approximate skyline calculation on big data

2021 ◽  
Author(s):  
Xingxing Xiao ◽  
Jianzhong Li
Keyword(s):  
Exact Solution ◽  
Big Data ◽  
Approximate Solution ◽  
Error Analysis ◽  
Linear Time ◽  
Approximate Algorithms ◽  
Skyline Query ◽  
Fixed Size ◽  
Skyline Queries ◽  
Input Size

Nowadays, big data is coming to the force in a lot of applications. Processing a skyline query on big data in more than linear time is by far too expensive and often even linear time may be too slow. It is obviously not possible to compute an exact solution to a skyline query in sublinear time, since an exact solution may itself have linear size. Fortunately, in many situations, a fast approximate solution is more useful than a slower exact solution. This paper proposes two sampling-based approximate algorithms for processing skyline queries. The first algorithm obtains a fixed size sample and computes the approximate skyline on it. The error of the algorithm is not only relatively small in most cases, but also is almost unaffected by the input size. The second algorithm returns an [Formula: see text]-approximation for the exact skyline efficiently. The running time of the algorithm has nothing to do with the input size in practical, achieving the goal of sublinearity on big data. Experiments verify the error analysis of the first algorithm, and show that the second is much faster than the existing skyline algorithms.

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