Numerical Solution of Fuzzy Differential Equations and its Applications

2014 ◽  
pp. 127-149 ◽  
Author(s):  
S. Chakraverty ◽  
Smita Tapaswini
Keyword(s):  
Differential Equations ◽  
Reaction Diffusion ◽  
Dynamical Behaviour ◽  
Reaction Diffusion Equation ◽  
Fuzzy Arithmetic ◽  
Initial Condition ◽  
Numerical Techniques ◽  
Science And Engineering ◽  
Fire Propagation ◽  
Reaction Diffusion Model

Theory of fuzzy differential equations is the important new developments to model various science and engineering problems of uncertain nature because this theory represents a natural way to model dynamical systems under uncertainty. Since, it is too difficult to obtain the exact solution of fuzzy differential equations so one may need reliable and efficient numerical techniques for the solution of fuzzy differential equations. In this chapter we have presented various numerical techniques viz. Euler and improved Euler type methods and Homotopy Perturbation Method (HPM) to solve fuzzy differential equations. Also application problems such as fuzzy continuum reaction diffusion model to analyse the dynamical behaviour of the fire with fuzzy initial condition is investigated. To analyse the fire propagation, the complex fuzzy arithmetic and computation are used to solve hyperbolic reaction diffusion equation. This analysis finds the rate of burning number of trees in bounds where wave variable/ time are defined in terms of fuzzy. Obtained results are compared with the existing solution to show the efficiency of the applied methods.

scholarly journals A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1653
Author(s):  
Tayyaba Akram ◽  
Muhammad Abbas ◽  
Ajmal Ali ◽  
Azhar Iqbal ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Technique ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Singular Kernel ◽  
Stability And Convergence ◽  
Theoretical Interpretation ◽  
Numerical Techniques ◽  
B Spline ◽  
Reaction Diffusion Model ◽  
Fractional Reaction

The time–fractional reaction–diffusion (TFRD) model has broad physical perspectives and theoretical interpretation, and its numerical techniques are of significant conceptual and applied importance. A numerical technique is constructed for the solution of the TFRD model with the non-singular kernel. The Caputo–Fabrizio operator is applied for the discretization of time levels while the extended cubic B-spline (ECBS) function is applied for the space direction. The ECBS function preserves geometrical invariability, convex hull and symmetry property. Unconditional stability and convergence analysis are also proved. The projected numerical method is tested on two numerical examples. The theoretical and numerical results demonstrate that the order of convergence of 2 in time and space directions.

Pattern Formation and Oscillations in Reaction–Diffusion Model with p53-Mdm2 Feedback Loop

2019 ◽  
Vol 29 (14) ◽  
pp. 1930040 ◽  
Author(s):  
Qianqian Zheng ◽  
Jianwei Shen ◽  
Zhijie Wang
Keyword(s):  
Pattern Formation ◽  
Feedback Loop ◽  
Reaction Diffusion ◽  
Vital Role ◽  
Reaction Diffusion Equation ◽  
Biological Mechanism ◽  
Reaction Diffusion Model ◽  
The Impact ◽  
Theoretical Results ◽  
Diffusive System

P53 plays a vital role in DNA repair, and several mathematical models of the p53-Mdm2 feedback loop were used to explain the biological mechanism. In this paper, a p53-Mdm2 model described by a delay reaction–diffusion equation is studied both analytically and numerically. This research aims to provide an understanding of the impact of delay and sustained pressure on the p53-Mdm2 dynamics and tries to explain some biological mechanism. It is found that the type of pattern formation is affected by Hopf bifurcation. Also, the amplitude equation in delay diffusive system is derived and it is shown that sustained stress plays an essential role in the function of p53. Finally, simulation is used to verify the theoretical results.

scholarly journals Modelling wildland fire propagation by tracking random fronts

2014 ◽  
Vol 14 (8) ◽  
pp. 2249-2263 ◽  
Author(s):  
G. Pagnini ◽  
A. Mentrelli
Keyword(s):  
Diffusion Equation ◽  
Level Set Method ◽  
Level Set ◽  
Wildland Fire ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Fire Propagation ◽  
Rate Of Spread ◽  
Hot Air ◽  
Break Zone

Abstract. Wildland fire propagation is studied in the literature by two alternative approaches, namely the reaction–diffusion equation and the level-set method. These two approaches are considered alternatives to each other because the solution of the reaction–diffusion equation is generally a continuous smooth function that has an exponential decay, and it is not zero in an infinite domain, while the level-set method, which is a front tracking technique, generates a sharp function that is not zero inside a compact domain. However, these two approaches can indeed be considered complementary and reconciled. Turbulent hot-air transport and fire spotting are phenomena with a random nature and they are extremely important in wildland fire propagation. Consequently, the fire front gets a random character, too; hence, a tracking method for random fronts is needed. In particular, the level-set contour is randomised here according to the probability density function of the interface particle displacement. Actually, when the level-set method is developed for tracking a front interface with a random motion, the resulting averaged process emerges to be governed by an evolution equation of the reaction–diffusion type. In this reconciled approach, the rate of spread of the fire keeps the same key and characterising role that is typical of the level-set approach. The resulting model emerges to be suitable for simulating effects due to turbulent convection, such as fire flank and backing fire, the faster fire spread being because of the actions by hot-air pre-heating and by ember landing, and also due to the fire overcoming a fire-break zone, which is a case not resolved by models based on the level-set method. Moreover, from the proposed formulation, a correction follows for the formula of the rate of spread which is due to the mean jump length of firebrands in the downwind direction for the leeward sector of the fireline contour. The presented study constitutes a proof of concept, and it needs to be subjected to a future validation.

scholarly journals The Uniqueness of Stationary Solution for Nonlinear Random Reaction-diffusion Equation

2020 ◽  
pp. 103-110
Author(s):  
Sofije Hoxha ◽  
Fejzi Kolaneci
Keyword(s):  
Invariant Measure ◽  
Diffusion Equation ◽  
Banach Spaces ◽  
Stationary Solution ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Reaction Diffusion Equation ◽  
Time Shift ◽  
Initial Condition ◽  
Real Noise

We study a nonlinear random reaction-diffusion problem in abstract Banach spaces, driven by a real noise, with random diffusion coefficient and random initial condition. We consider a polynomial non linear term. The reaction-diffusion equation belongs to the class of parabolic stochastic partial differential equations. We assume that the initial condition is an element of Hilbert space. The real noise is a Wiener process. We construct a suitable stochastic basis and define the solution of reaction-diffusion problem in the weak sense. We define the stationary process in abstract Banach spaces in the strong sense of Doob-Rozanov. That is, the probability density function of the stochastic process is independent of time shift. We define the invariant measure for random reaction-diffusion equation in the sense of Arnold, DaPrato, and Zabczyk [1,2]. In other words, we define the invariant measure for random dynamical system, associated with random reaction-diffusion problem. Using the Variation Inequalities Theory, we prove the uniqueness of stationary solution for nonlinear random reaction-diffusion problem. The obtained theoretical results have several applications in Quantum Physics, Biology, Medicine, and Economic Sciences. Especially, we can study the existence of stationary solution for the stochastic models of tumor growth.

scholarly journals An ancient Chinese algorithm for two-point boundary problems and its application to the Michaelis-Menten kinetics

2021 ◽  
Vol 1 (4) ◽  
pp. 172-176
Author(s):  
Ji-Huan He ◽  
◽  
Shuai-Jia Kou ◽  
Hamid M. Sedighi ◽  
◽  
...  
Keyword(s):  
Infinite Series ◽  
Solution Process ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Initial Condition ◽  
Point Boundary ◽  
Boundary Problems ◽  
Nonlinear Reaction ◽  
Nonlinear Reaction Diffusion Equation ◽  
Taylor Series Method

<abstract><p>Taylor series method is simple, and an infinite series converges to the exact solution for initial condition problems. For the two-point boundary problems, the infinite series has to be truncated to incorporate the boundary conditions, making it restrictively applicable. Here is recommended an ancient Chinese algorithm called as <italic>Ying Buzu Shu</italic>, and a nonlinear reaction diffusion equation with a Michaelis-Menten potential is used as an example to show the solution process.</p></abstract>

scholarly journals New Exact Solutions of the Brusselator Reaction Diffusion Model Using the Exp-Function Method

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
F. Khani ◽  
F. Samadi ◽  
S. Hamedi-Nezhad
Keyword(s):  
Exact Solutions ◽  
Diffusion Model ◽  
Function Method ◽  
Soliton Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
New Exact Solutions ◽  
Nonlinear Reaction ◽  
Nonlinear Reaction Diffusion Equation ◽  
Reaction Diffusion Model

Firstly, using a series of transformations, the Brusselator reaction diffusion model is reduced into a nonlinear reaction diffusion equation, and then through using Exp-function method, more new exact solutions are found which contain soliton solutions. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The results show the reliability and efficiency of the proposed method.

A scaling transformation method and exact solutions of nonlinear reaction–diffusion model

2020 ◽  
Vol 34 (31) ◽  
pp. 2050356
Author(s):  
Xin Wang ◽  
Yang Liu
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Integral Form ◽  
Transformation Method ◽  
Variable Coefficients ◽  
Reaction Diffusion Equation ◽  
Polynomial Method ◽  
Nonlinear Odes ◽  
Reaction Diffusion Model ◽  
Scaling Transformation

We find a very simple scaling transformation method by which a kind of rank non-homogenous second order nonlinear ODEs is reduced to an integral form. Comparing with the canonical-like transformation method and the first integral method, our method provides a simple mechanism of reduction of such equations. As an application, a reaction–diffusion equation with variable coefficients is integrated, and its exact cnoidal wave solution is obtained. Furthermore, by using the complete discrimination system for polynomial method, more exact solutions are obtained.

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