scholarly journals Exact analytical solutions of fractional order telegraph equations via triple Laplace transform

2018 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Rahmat Ali Khan ◽  
◽  
Yongjin Li ◽  
Fahd Jarad ◽  
◽  
...  
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Exact Analytical Solutions ◽  
Telegraph Equations

On the approximate inverse Laplace transform of the transfer function with a single fractional order

2020 ◽  
pp. 014233122097766
Author(s):  
Ali Yüce ◽  
Nusret Tan
Keyword(s):  
Transfer Function ◽  
Laplace Transform ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Transfer Functions ◽  
Inverse Laplace Transform ◽  
Fractional Order Systems ◽  
Approximate Inverse ◽  
Classical Mathematics ◽  
Curve Fitting Method

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of [Formula: see text]. Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for [Formula: see text]. By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of [Formula: see text] for [Formula: see text]. Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.

scholarly journals Exact Analytical Solutions of Linear Dissipative Wave Equations via Laplace Transform Method

2021 ◽  
pp. 425-434
Author(s):  
Muhammad Jamil ◽  
Rahmat Ali Khan ◽  
Kamal Shah
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Analytical Solutions ◽  
Wave Equations ◽  
Two Dimensional ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Exact Analytical Solutions ◽  
Double Laplace Transform ◽  
Wave Phenomena

A wave phenomena evolved day after day, as various concepts regarding waves appeared with the passage of time. These phenomena are generally modelled mathematically by partial differential equations (PDEs). In this research, we investigate the exact analytical solutions of one and two dimensional linear dissipative wave equations which are modelled by second order PDEs with use of some initial and boundary conditions. We use double Laplace transform (DLT) and triple Laplace transform (TLT) methods to determine these exact analytical solutions. We provide examples with figures to test effectiveness of this scheme of Laplace transform

scholarly journals Laplace Transform Collocation Method for Solving Hyperbolic Telegraph Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Adebayo O. Adewumi ◽  
Saheed O. Akindeinde ◽  
Adebayo A. Aderogba ◽  
Babatunde S. Ogundare
Keyword(s):  
Laplace Transform ◽  
Collocation Method ◽  
Analytical Solutions ◽  
Trial Function ◽  
Numerical Scheme ◽  
Telegraph Equation ◽  
One Dimensional ◽  
Laplace Transform Technique ◽  
Telegraph Equations ◽  
New Form

This article presents a new numerical scheme to approximate the solution of one-dimensional telegraph equations. With the use of Laplace transform technique, a new form of trial function from the original equation is obtained. The unknown coefficients in the trial functions are determined using collocation method. The efficiency of the new scheme is demonstrated with examples and the approximations are in excellent agreement with the analytical solutions. This method produced better approximations than the ones produced with the standard weighted residual methods.

Fractional order generalized thermoelastic response in a half space due to a periodically varying heat source

2018 ◽  
Vol 14 (1) ◽  
pp. 2-15 ◽  
Author(s):  
Jitesh Tripathi ◽  
Shrikant Warbhe ◽  
K.C. Deshmukh ◽  
Jyoti Verma
Keyword(s):  
Mathematical Model ◽  
Heat Source ◽  
Laplace Transform ◽  
Half Space ◽  
Fractional Order ◽  
Order Parameter ◽  
Order Theory ◽  
Numerical Inversion ◽  
Content Type ◽  
Copper Material

Purpose The present work is concerned with the solution of a fractional-order thermoelastic problem of a two-dimensional infinite half space under axisymmetric distributions in which lower surface is traction free and subjected to a periodically varying heat source. The thermoelastic displacement, stresses and temperature are determined within the context of fractional-order thermoelastic theory. To observe the variations of displacement, temperature and stress inside the half space, the authors compute the numerical values of the field variables for copper material by utilizing Gaver-Stehfast algorithm for numerical inversion of Laplace transform. The effects of fractional-order parameter on the variations of field variables inside the medium are analyzed graphically. The paper aims to discuss these issues. Design/methodology/approach Integral transform technique and Gaver-Stehfast algorithm are applied to prepare the mathematical model by considering the periodically varying heat source in cylindrical co-ordinates. Findings This paper studies a problem on thermoelastic interactions in an isotropic and homogeneous elastic medium under fractional-order theory of thermoelasticity proposed by Sherief (Ezzat and El-Karamany, 2011b). The analytic solutions are found in Laplace transform domain. Gaver-Stehfast algorithm (Ezzat and El-Karamany, 2011d; Ezzat, 2012; Ezzat, El Karamany, Ezzat, 2012) is used for numerical inversion of the Laplace transform. All the integrals were evaluated using Romberg’s integration technique (El-Karamany et al., 2011) with variable step size. A mathematical model is prepared for copper material and the results are presented graphically with the discussion on the effects of fractional-order parameter. Research limitations/implications Constructed purely on theoretical mathematical model by considering different parameters and the functions. Practical implications The system of equations in this paper may prove to be useful in studying the thermal characteristics of various bodies in real-life engineering problems by considering the time fractional derivative in the field equations. Originality/value In this problem, the authors have used the time fractional-order theory of thermoelasticity to solve the problem for a half space with a periodically varying heat source to control the speed of wave propagation in terms of heat and elastic waves for different conductivity like weak conductivity, moderate conductivity and super conductivity which is a new and novel contribution.

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