The time fractional order differential equations are fundamental tools that are used for modeling neuronal dynamics. These equations are obtained by substituting the time derivative of order where , in the standard equation with the Caputo fractional formula. In this paper, two implicit difference schemes: the linearly Euler implicit and the Crank-Nicolson (CN) finite difference schemes, are employed in solving a one-dimensional time-fractional semilinear equation with Dirichlet boundary conditions. Moreover, the consistency, stability and convergence of the proposed schemes are investigated. We prove that the IEM is unconditionally stable, while CNM is conditionally stable. Furthermore, a comparative study between these two schemes will be conducted via numerical experiments. The efficiency of the proposed schemes in terms of absolute errors, order of accuracy and computing time will be reported and discussed.

On The Solutions of Wave Equation in ThreeDimensions using D'alembert Formula

Recent Weather Observation in South - West Asia Region had shown an increase in both frequency and intensity of Dust phenomena. The increasing trend of Dust phenomena was used to view the future outlook for dust phenomena. Locating Dust Storm Sources (DSS) in the region. Using the backward - projection and Remot-Sensing method DSS were used. Also, mineralogical study of dust particles was done to give an idea about their sources. It is seen that silt is the major constituent of the collected samples with percentage of 42.6% followed immediately by clay 40.8%, and sand is the minor constituent 5.8%. Chemical analysis of dust specimens showed that oxides of Silicon, Aluminum, Calcium, Ferrous and Magnesium are the major constituents of accumulated dust with minor constituent of Potassium, Sodium and Titanium oxides and some Sulphates. Shapes of dust were mainly irregular plates, while sand are spherical and sometimes oval.

This paper is concerned with the blow-up properties for a semilinear heat equation with homogenous Dirichlet boundary conditions, defined on a ball in Rⁿ . We show that the blow-up in this problem can occur in finite time at only a single point. Moreover, the upper blow-up rate estimate for this problem is derived.

This paper is concerned with the numerical blow-up time for a coupled system of two one-dimensional semilinear parabolic equations with zero Dirichlet boundary conditions. Firstly, we derive the semi-discrete problem and prove that the blow-up solution and numerical blow-up time of the semi-discrete problem are convergent to the theoretical ones, as we refine the space-time grids. Secondly, we derive two fully discrete formulas of standard finite difference methods: the explicit Euler and implicit Euler schemes, with non-fixed time-stepping formula. In addition, we investigate the consistency, stability and convergence of the proposed schemes. Finally, we conduct two numerical experiments to show the accuracy and efficiency of the proposed schemes. Namely, for each experiment, we use the proposed schemes to calculate the numerical blow-up time, error bounds and the numerical order of convergence for blow-up times.

This paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diffusion equations with gradient terms. The main goal is to study the influence of the gradient terms on the blow-up profile. Namely, under some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.