This paper is devoted to the study of an eigenvalue second order differential EQUATION, supplied with homogenous Dirichlet conditions and set on the real line. In the linear case, the EQUATION arises in the study of a reaction-diffusion system involved in disease propagation throughout a given population. Under some relations upon the real parameters and coefficients, we present some existence and nonexistence results. We use a variational method and fixed point arguments.

In this article, we propose a method for the solution of the GENERALIZED Burger– FISHER EQUATION. The method is developed using CAS wavelets in conjunction with quasi-linearization technique. The operational matrices for the CAS wavelets are derived and constructed. Error analysis and procedure of implementation of the method are provided. We compare the results produce by present method with some well known results and show that the present method is more accurate, efficient, and applicable.

In the present work, an efficient numerical technique, called q-homotopy analysis transform method (briefly, q-HATM), is applied to nonlinear FISHER’ s EQUATION of fractional order. The homotopy polynomials are employed, in order to handle the nonlinear terms. Numerical examples are illustrated to examine the efficiency of the proposed technique. The suggested algorithm provides the auxiliary parameters  and n, which help us to control and adjust the convergence region of the series solution. The outcomes of the study reveal that the q-HATM is computationally very effective and accurate to analyse nonlinear fractional differential EQUATIONs.

In this work, we investigate a fractional version of the FISHER EQUATION and solve it by using an efficient iteration technique based on the Haar wavelet operational matrices. In fact, we convert the nonlinear EQUATION into a Sylvester EQUATION by the Haar wavelet collocation iteration method (HWCIM) to obtain the solution. We provide four numerical examples to illustrate the simplicity and efficiency of the technique.

In this paper, we are interested to study the Sine-Gordon EQUATION in GENERALIZED function theory, we give a result of existence and uniqueness of GENERALIZED solution with initial data are distributions (elements of the Colombeau algebra). Then we study the association concept with the classical solution.

A nonstandard finite-difference (NSFD) scheme for FISHER’ s EQUATION by using Richtmyer’ s (3, 1, 1) implicit formula has been presented, in this work. On nonstandard finite-difference scheme, two special cases of Richtmyer formula have been applied. The suitable functions in the denominator fraction of our NSFD scheme have been replaced to guarantee the highly accurate of the approximation. Furthermore, the analyses of stability, convergence, consistency for the NSFD method, have been provided. By calculating the absolute error, the comparison of these methods has been presented in some Examples and the results have shown that the error of our NSFD scheme is lower than the others. Finally, a comparison of these methods and the differential quadrature method (DQM) to solve the FISHER’ s EQUATION reveals that these techniques work better and give highly accurate results.

The Jump-Diffusion EQUATION is a generalization of the Langevin EQUATION; it has been usually applied to reconstruct discontinuous stochastic processes. In this article, by using this EQUATION, we investigate the electrocardiogram of the electric activity of the heart beat, for three groups of subjects with normal, atrial fibrillation and ventricular arrhythmia. At first, we demonstrate that the time series of electrocardiogram is a discontinuous process that can be modeled by the jump-diffusion EQUATION. Then, by calculating the Kramers-Moyal coefficients related to this EQUATION, we show that there is a significant difference between the heart dynamics of the normal subjects and the ones with heart failure exists. Finally, we introduce a measure that may be used for the diagnosis of heart failures.