scholarly journals Numerical Solution of Fractional Order Anomalous Subdiffusion Problems Using Radial Kernels and Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Muhammad Taufiq ◽  
Marjan Uddin
Keyword(s):  
Laplace Transform ◽  
Numerical Solution ◽  
Fractional Order ◽  
Finite Differences ◽  
Numerical Scheme ◽  
Inverse Laplace Transform ◽  
Fractional Operator ◽  
Integration Technique ◽  
High Dimensions ◽  
Time Sensitivity

By coupling of radial kernels and localized Laplace transform, a numerical scheme for the approximation of time fractional anomalous subdiffusion problems is presented. The fractional order operators are well suited to handle by Laplace transform and radial kernels are also built for high dimensions. The numerical computations of inverse Laplace transform are carried out by contour integration technique. The computation can be done in parallel and no time sensitivity is involved in approximating the time fractional operator as contrary to finite differences. The proposed numerical scheme is stable and accurate.

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 Parameterizing Generalized Laguerre Functions to Compute the Inverse Laplace Transform of Fractional Order Transfer Functions

MENDEL ◽  
2018 ◽  
Vol 24 (1) ◽  
pp. 79-84
Author(s):  
Vilem Karsky
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Transfer Functions ◽  
Operator Space ◽  
Inverse Laplace Transform ◽  
Laguerre Functions ◽  
Two Systems ◽  
Novel Method

This article concentrates on using generalized Laguerre functions to compute the inverse Laplace transform of fractional order transfer functions. A novel method for selecting the timescale parameter of generalized Laguerre functions in the operator space is introduced and demonstrated on two systems with fractional order transfer functions.

scholarly journals Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Xiaoli Qiang ◽  
Kamran ◽  
Abid Mahboob ◽  
Yu-Ming Chu
Keyword(s):  
Numerical Methods ◽  
Laplace Transform ◽  
Fractional Order ◽  
Integrodifferential Equations ◽  
Inverse Laplace Transform ◽  
Laplace Domain ◽  
The Real ◽  
Volterra Integrodifferential Equation ◽  
Engineering Sciences ◽  
Real Domain

Laplace transform is a powerful tool for solving differential and integrodifferential equations in engineering sciences. The use of Laplace transform for the solution of differential or integrodifferential equations sometimes leads to the solutions in the Laplace domain that cannot be inverted to the real domain by analytic methods. Therefore, we need numerical methods to invert the solution to the real domain. In this work, we construct numerical schemes based on Laplace transform for the solution of fractional-order Volterra integrodifferential equations in the sense of the Atangana-Baleanu Caputo derivative. We propose two numerical methods for approximating the solution of fractional-order linear and nonlinear Volterra integrodifferential equations. In our scheme, the inverse Laplace transform is approximated using a contour integration method and Stehfest method. Some numerical experiments are performed to check the accuracy and efficiency of the methods. The results obtained using these methods are compared.

scholarly journals Fractional-Order Controller Design for Oscillatory Fractional Time-Delay Systems Based on the Numerical Inverse Laplace Transform Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Lu Liu ◽  
Feng Pan ◽  
Dingyu Xue
Keyword(s):  
Time Delay ◽  
Laplace Transform ◽  
Fractional Order ◽  
Control Method ◽  
Delay System ◽  
Inverse Laplace Transform ◽  
Delay Systems ◽  
Time Delay System ◽  
Numerical Inverse Laplace Transform ◽  
Fractional Order Controller

Fractional-order time-delay system is thought to be a kind of oscillatory complex system which could not be controlled efficaciously so far because it does not have an analytical solution when using inverse Laplace transform. In this paper, a type of fractional-order controller based on numerical inverse Laplace transform algorithm INVLAP was proposed for the mentioned systems by searching for the optimal controller parameters with the objective function of ITAE index due to the verified nature that fractional-order controllers were the best means of controlling fractional-order systems. Simulations of step unit tracking and load-disturbance responses of the proposed fractional-order optimalPIλDμcontroller (FOPID) and corresponding conventional optimal PID (OPID) controller have been done on three typical kinds of fractional time-delay system with different ratio between time delay (L) and time constant (T) and a complex high-order fractional time delay system to verify the availability of the presented control method.

scholarly journals A numerical and analytical study of SE(Is)(Ih)AR epidemic fractional order COVID-19 model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hasib Khan ◽  
Razia Begum ◽  
Thabet Abdeljawad ◽  
M. Motawi Khashan
Keyword(s):  
Mathematical Model ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Analytical Study ◽  
Existence Results ◽  
Fractional Operator ◽  
Virus Spread ◽  
Fractional Order Derivative ◽  
The Given

AbstractThis article describes the corona virus spread in a population under certain assumptions with the help of a fractional order mathematical model. The fractional order derivative is the well-known fractal fractional operator. We have given the existence results and numerical simulations with the help of the given data in the literature. Our results show similar behavior as the classical order ones. This characteristic shows the applicability and usefulness of the derivative and our numerical scheme.

scholarly journals Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Jian Wang ◽  
Kamran ◽  
Ayesha Jamal ◽  
Xuemei Li
Keyword(s):  
Laplace Transform ◽  
Integrodifferential Equation ◽  
Numerical Scheme ◽  
High Accuracy ◽  
Trapezoidal Rule ◽  
Inverse Laplace Transform ◽  
Fredholm Type ◽  
The Laplace Transform ◽  
Linear And Nonlinear ◽  
The Given

In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

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