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 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 (Kink; Kink; Kink; Kink) and (Pulse; Pulse; Pulse; Pulse) Solutions of a Set of Four Equations Modeled in a Nonlinear Hybrid Electrical Line with Crosslink Capacitor

2019 ◽  
pp. 1-14
Author(s):  
Tiague Takongmo Guy ◽  
Jean Roger Bogning
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Solitary Waves ◽  
Nonlinear Partial Differential Equations ◽  
Mathematical Methods ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
Hybrid Parts ◽  
Nonlinear Hybrid

The physics system that helps us in the study of this paper is a nonlinear hybrid electrical line with crosslink capacitor. Meaning it is composed of two different nonlinear hybrid parts Linked by capacitors with identical constant capacitance. We apply Kirchhoff laws to the circuit of the line to obtain new set of four nonlinear partial differential equations which describe the simultaneous dynamics of four solitary waves. Furthermore, we apply efficient mathematical methods based on the identification of coefficients of basic hyperbolic functions to construct exact solutions of those set of four nonlinear partial differential equations. The obtained results have enabled us to discover that, one of the two nonlinear hybrid electrical line with crosslink capacitor that we have modeled accepts the simultaneous propagation of a set of four solitary waves of type (Pulse; Pulse; Pulse; Pulse), while the other accepts the simultaneous propagation of a set of four solitary waves of type (Kink; Kink; Kink; Kink) when certain conditions we have established are respected. We ameliorate the quality of the signals by changing the sinusoidal waves that are supposed to propagate in the hybrid electrical lines with crosslink capacitor to solitary waves which are propagating in the new nonlinear hybrid electrical lines; we therefore, facilitate the choice of the type of line relative to the type of signal that we want to transmit.

Oscillation Criteria for Semilinear Equations in General Domains

1976 ◽  
Vol 19 (2) ◽  
pp. 137-144 ◽  
Author(s):  
W. Allegretto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Linear Equations ◽  
Oscillation Theory ◽  
General Nature ◽  
Nonlinear Ordinary Differential Equations ◽  
Exterior Domains ◽  
Semilinear Equations

Oscillation theory for nonlinear ordinary differential equations has been extensively developed in recent years by several authors. We refer the reader to the recent paper by Noussair and Swanson, [2], where an extensive bibliography may be found. The situation is somewhat different for the case of second order partial differential equations, an area which has recently been virtually untouched, except for the establishment of criteria which depend on a comparison with suitable linear equations and therefore are essentially linear in nature. A bibliography of such results may also be found in [2]. Of a more general nature have been the paper by the author [1], and the more recent work of Noussair and Swanson, [2]. Although the methods employed and the equations considered in [1] and [2] are different, both papers obtained results only for a class of exterior domains of Rn, of which the typical example is the complement of a bounded sphere.

Modified wavelet method for solving fractional variational problems

2020 ◽  
pp. 107754632093202
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani ◽  
Mohsen Razzaghi
Keyword(s):  
Error Analysis ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variational Problems ◽  
High Accuracy ◽  
Construction Method ◽  
Computational Method ◽  
Wavelet Method ◽  
Fractional Variational Problems ◽  
Operational Matrix Of Integration

In this article, a newly modified Bessel wavelet method for solving fractional variational problems is considered. The modified operational matrix of integration based on Bessel wavelet functions is proposed for solving the problems. In the process of computing this matrix, we have tried to provide a high-accuracy operational matrix. We also introduce the pseudo-operational matrix of derivative and the dual operational matrix with the coefficient. Also, we investigate the error analysis of the computational method. In the examples section, the behavior of the approximate solutions with respect to various parameters involved in the construction method is tested to illustrate the efficiency and accuracy of the proposed method.

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