scholarly journals A Collocation Method for the Numerical Solution of Nonlinear Fractional Dynamical Systems

Algorithms ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 156 ◽  
Author(s):  
Francesca Pitolli
Keyword(s):  
Applied Sciences ◽  
Dynamical Systems ◽  
Numerical Methods ◽  
Numerical Solution ◽  
Collocation Method ◽  
Fractional Order ◽  
Time Derivative ◽  
Test Problems ◽  
Fractional Differential

Fractional differential problems are widely used in applied sciences. For this reason, there is a great interest in the construction of efficient numerical methods to approximate their solution. The aim of this paper is to describe in detail a collocation method suitable to approximate the solution of dynamical systems with time derivative of fractional order. We will highlight all the steps necessary to implement the corresponding algorithm and we will use it to solve some test problems. Two Mathematica Notebooks that can be used to solve these test problems are provided.

scholarly journals Numerical Solution of Partial Integrodifferential Equations of Diffusion Type

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Imran Aziz ◽  
Imran Khan
Keyword(s):  
Applied Sciences ◽  
Dynamical Systems ◽  
Numerical Solution ◽  
Numerical Method ◽  
Collocation Method ◽  
Numerical Results ◽  
Integrodifferential Equations ◽  
Benchmark Problems ◽  
Diffusion Type ◽  
One Dimensional

A collocation method based on linear Legendre multiwavelets is developed for numerical solution of one-dimensional parabolic partial integrodifferential equations of diffusion type. Such equations have numerous applications in many problems in the applied sciences to model dynamical systems. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency, and robustness of the proposed method.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

scholarly journals Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Imtiaz Wasim ◽  
Muhammad Abbas ◽  
Muhammad Amin
Keyword(s):  
Collocation Method ◽  
Numerical Technique ◽  
Time Derivative ◽  
Fourier Method ◽  
Spatial Dimension ◽  
Test Problems ◽  
Spline Collocation ◽  
Von Neumann ◽  
B Spline ◽  
Spline Collocation Method

In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. This technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. The scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. The numerical results are in good agreement with known exact solutions and the existing schemes in literature.

A New Result on Fractional Differential Inequality and Applications to Control of Dynamical Systems

2019 ◽  
Vol 141 (9) ◽  
Author(s):  
Ngo Van Hoa ◽  
Tran Minh Duc ◽  
Ho Vu
Keyword(s):  
Dynamical Systems ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Differential Inequality ◽  
Sliding Mode ◽  
Control Law ◽  
Asymptotically Stable ◽  
Mode Control ◽  
Fractional Differential

In this work, we establish a new estimate result for fractional differential inequality, and this inequality is used to derive a robust sliding mode control law for the fractional-order (FO) dynamic systems. The sliding mode control law is provided to make the states of the system asymptotically stable. Some examples are given to illustrate the results.

A new structure formulations for cubic B-spline collocation method in three and four-dimensions

2020 ◽  
Vol 9 (1) ◽  
pp. 432-448
Author(s):  
K. R. Raslan ◽  
Khalid K. Ali
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Collocation Method ◽  
Test Problems ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method ◽  
Four Dimensions ◽  
B Splines

AbstractIn this work, we introduce a new construct to the cubic B-spline collocation method in the three and four-dimensions. The cubic B-splines method format is displayed in one, two, three, and four-dimensions format. These constructions are of utmost importance in solving differential equations in their various dimensions, which have applications in many fields of science. The efficiency and accuracy of the proposed methods are demonstrated by its application to a few test problems in two, three, and four dimensions. Also, comparing the exact solutions and with the results obtained by using other numerical methods available in the literature as much as possible.

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