Numerical Solution of Nonlinear Space–Time Fractional-Order Advection–Reaction–Diffusion Equation

2020 ◽  
Vol 15 (6) ◽  
Author(s):  
Kushal Dhar Dwivedi ◽  
Rajeev ◽  
Subir Das ◽  
Dumitru Baleanu
Keyword(s):  
Fractional Order ◽  
Operational Matrix ◽  
Reaction Diffusion ◽  
Space Time ◽  
Reaction Diffusion Equation ◽  
Processing Unit ◽  
Algebraic Equations ◽  
Central Processing ◽  
Reaction Diffusion Model ◽  
Fibonacci Polynomial

Abstract In this article, a new algorithm is proposed to solve the nonlinear fractional-order one-dimensional solute transport system. The spectral collocation technique is considered with the Fibonacci polynomial as a basis function for the approximation. The Fibonacci polynomial is used to obtain derivative in terms of an operational matrix. The proposed algorithm is actually based on the fact that the terms of the considered problem are approximated through a series expansion of double Fibonacci polynomials and then collocated those on specific points, which provide a system of nonlinear algebraic equations which are solved by using Newton's method. To validate the precision of the proposed method, it is applied to solve three different problems having analytical solutions. The comparison of the results through error analysis is depicted through tables which clearly show the higher accuracy of order of convergence of the proposed method in less central processing unit (CPU) time. The salient feature of the article is the graphical exhibition of the movement of solute concentration for different particular cases due to the presence and absence of reaction term when the proposed scheme is applied to the considered nonlinear fractional-order space–time advection–reaction–diffusion model.

scholarly journals Two-dimensional nonlinear time fractional reaction–diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media

Meccanica ◽  
2020 ◽  
Author(s):  
P. Pandey ◽  
S. Das ◽  
E-M. Craciun ◽  
T. Sadowski
Keyword(s):  
Diffusion Equation ◽  
Present Article ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Linear Algebraic Equations ◽  
Non Linear

AbstractIn the present article, an efficient operational matrix based on the famous Laguerre polynomials is applied for the numerical solution of two-dimensional non-linear time fractional order reaction–diffusion equation. An operational matrix is constructed for fractional order differentiation and this operational matrix converts our proposed model into a system of non-linear algebraic equations through collocation which can be solved by using the Newton Iteration method. Assuming the surface layers are thermodynamically variant under some specified conditions, many insights and properties are deduced e.g., nonlocal diffusion equations and mass conservation of the binary species which are relevant to many engineering and physical problems. The salient features of present manuscript are finding the convergence analysis of the proposed scheme and also the validation and the exhibitions of effectiveness of the method using the order of convergence through the error analysis between the numerical solutions applying the proposed method and the analytical results for two existing problems. The prominent feature of the present article is the graphical presentations of the effect of reaction term on the behavior of solute profile of the considered model for different particular cases.

Formalism for discrete multidimensional dynamic systems

Kybernetes ◽  
2016 ◽  
Vol 45 (10) ◽  
pp. 1555-1575
Author(s):  
Joan C. Micó ◽  
Antonio Caselles ◽  
David Soler ◽  
Pantaleón D. Romero
Keyword(s):  
Dynamic System ◽  
Social Systems ◽  
Urban System ◽  
Reaction Diffusion ◽  
Space Time ◽  
Reaction Diffusion Equation ◽  
Content Type ◽  
Time Dynamic ◽  
Reaction Diffusion Model ◽  
Scope Of Application

Purpose The purpose of this paper is to suggest a formalism given by an equation suitable for simulating discrete systems with space-time variation in addition to other change variables. With such formalism, multidimensional dynamical models of discrete complex systems, such as the social systems and ecosystems, can be built. Design/methodology/approach This formalism is named as discrete multidimensional dynamic system (DMDS). The DMDS provides a way to consider the variation of the density of a state variable with regard to the variables of the change space as a function of multidimensional rates. Multidimensional rates describe this evolution as a consequence of the relation of each multidimensional-point with a given set of other points of the change space. This relation contains the accessibility domains (sets of space points with which each space point is related). Findings This equation is compared with both the reaction-diffusion equation written in its finite difference form and the cellular-automata model, demonstrating its compatibility with them and an increase in generality, widening the scope of application. The steps to construct models of systems with multidimensional variation based on the equation that defines the DMDS are specified and tested. Research limitations/implications Through the DMDS and a well-stated methodology, an application case is provided in order to describe the multidimensional demographic dynamics of an urban system. In this case, the numerical evolution of the population density by districts and cohorts is determined by the DMDS based on some hypothesis about functions of population diffusion between the different districts of the system. Originality/value The scope of application of the space-time dynamic system (STDS), given by the authors in a previous work, has been extended to discrete and multidimensional systems. STDS model produces better results than the reaction-diffusion model in validation.

scholarly journals Numerical Analysis WSGD Scheme for One- and Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation with Collocation Method via Fractional B-Spline

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mohammad Ramezani
Keyword(s):  
Diffusion Equation ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
B Spline ◽  
Computational Work ◽  
Distributed Order ◽  
Fractional Reaction

AbstractThe main propose of this paper is presenting an efficient numerical scheme to solve WSGD scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation. The proposed method is based on fractional B-spline basics in collocation method which involve Caputo-type fractional derivatives for $$0 < \alpha < 1$$ 0 < α < 1 . The most significant privilege of proposed method is efficient and quite accurate and it requires relatively less computational work. The solution of consideration problem is transmute to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. The finally, several numerical WSGD Scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation.

scholarly journals A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Closed Form Solutions ◽  
Homotopy Perturbation ◽  
In Series ◽  
The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

scholarly journals An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1755
Author(s):  
M. S. Al-Sharif ◽  
A. I. Ahmed ◽  
M. S. Salim
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Fractional Differential

Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

An efficient approximate method for solving two-dimensional fractional optimal control problems using generalized fractional order of Bernstein functions

2020 ◽  
Author(s):  
Ali Ketabdari ◽  
Mohammad Hadi Farahi ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Fractional Order ◽  
Operational Matrix ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Bernstein Functions ◽  
Fractional Optimal Control Problems ◽  
And Control

Abstract We define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

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