scholarly journals Laplace-SBA Method for Solving Nonlinear Coupled Burger's Equations

2021 ◽  
Vol 14 (3) ◽  
pp. 842-862
Author(s):  
Joseph Bonazebi-Yindoula
Keyword(s):  
Fluid Dynamics ◽  
Exact Solution ◽  
Numerical Methods ◽  
Laplace Transform ◽  
Exact Solutions ◽  
Partial Derivative ◽  
Basic Principles ◽  
The Laplace Transform ◽  
Fluid Dynamics Equations ◽  
Burger's Equations

Burger’s equations, an extension of fluid dynamics equations, are typically solved by several numerical methods. In this article, the laplace-Somé Blaise Abbo method is used to solve nonlinear Burger equations. This method is based on the combination of the laplace transform and the SBA method. After reminders of the laplace transform, the basic principles of the SBA method are described. The process of calculating the Laplace-SBA algorithm for determining the exact solution of a linear or nonlinear partial derivative equation is shown. Thus, three examplesof PDE are solved by this method, which all lead to exact solutions. Our results suggest that this method can be extended to other more complex PDEs.

Some further results of the laplace transform for variable–order fractional difference equations

2019 ◽  
Vol 22 (6) ◽  
pp. 1641-1654 ◽  
Author(s):  
Dumitru Baleanu ◽  
Guo–Cheng Wu
Keyword(s):  
Laplace Transform ◽  
Exact Solutions ◽  
Difference Equations ◽  
Response Analysis ◽  
Variable Order ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
The Laplace Transform ◽  
Linear Differential ◽  
Time Systems

Abstract The Laplace transform is important for exact solutions of linear differential equations and frequency response analysis methods. In comparison with the continuous–time systems, less results can be available for fractional difference equations. This study provides some fundamental results of two kinds of fractional difference equations by use of the Laplace transform. Some discrete Mittag–Leffler functions are defined and their Laplace transforms are given. Furthermore, a class of variable–order and short memory linear fractional difference equations are proposed and the exact solutions are obtained.

scholarly journals Exact Solutions of Fragmentation Equations with General Fragmentation Rates and Separable Particles Distribution Kernels

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
S. C. Oukouomi Noutchie ◽  
E. F. Doungmo Goufo
Keyword(s):  
Exact Solution ◽  
Evolution Equation ◽  
Laplace Transform ◽  
Exact Solutions ◽  
Method Of Characteristics ◽  
Open Problems ◽  
The Method Of Characteristics ◽  
Sudden Appearance ◽  
Many Particles

We make use of Laplace transform techniques and the method of characteristics to solve fragmentation equations explicitly. Our result is a breakthrough in the analysis of pure fragmentation equations as this is the first instance where an exact solution is provided for the fragmentation evolution equation with general fragmentation rates. This paper is the key for resolving most of the open problems in fragmentation theory including “shattering” and the sudden appearance of infinitely many particles in some systems with initial finite particles number.

scholarly journals Rigorous time domain responses of polarizable media II

1998 ◽  
Vol 41 (3) ◽  
Author(s):  
M. Caputo ◽  
W. Plastino
Keyword(s):  
Exact Solution ◽  
Laplace Transform ◽  
Time Domain ◽  
Input Data ◽  
Synthetic Data ◽  
Induced Polarization ◽  
Discharge Data ◽  
Time Domain Data ◽  
Previous Note ◽  
The Laplace Transform

We present and test in detail with synthetic data a method which may be used to retrieve the parameters describing the induced polarization properties of media which fit the generally accepted frequency dependent formula of Cole and Cole (1941) (CC model). We use time domain data and rigorous formulae obtained from the exact solution of the problem found in a previous note (Caputo, 1996). The observed data considered here are the theoretical responses of the medium to box inputs of given duration in media defined with different parameters; however, as is usually done, only the discharge data are used (Patella >F2<et al.>F1<, 1987). The curve at the beginning of the discharge is studied in some detail. The method is successful in identifying the parameters when the data fit the CC model; if the medium is not exactly of the CC type the method may also help identify how the medium departs from the CC model. The Laplace Transform of the discharge for a box type input data is also given.

scholarly journals Semianalytical Solutions of Some Nonlinear-Time Fractional Models Using Variational Iteration Laplace Transform Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Javed Iqbal ◽  
Khurram Shabbir ◽  
Liliana Guran
Keyword(s):  
Laplace Transform ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Fractional Partial Differential Equations ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Variational Iteration ◽  
The Laplace Transform ◽  
Fractional Models ◽  
Iteration Technique

In this work, we combined two techniques, the variational iteration technique and the Laplace transform method, in order to solve some nonlinear-time fractional partial differential equations. Although the exact solutions may exist, we introduced the technique VITM that approximates the solutions that are difficult to find. Even a single iteration best approximates the exact solutions. The fractional derivatives being used are in the Caputo-Fabrizio sense. The reliability and efficiency of this newly introduced method is discussed in details from its numerical results and their graphical approximations. Moreover, possible consequences of these results as an application of fixed-point theorem are placed before the experts as an open problem.

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