Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems

This paper deals with a class of Bratu’s type, Troesch’s and nonlocal elliptic boundary value problems. Due to strong nonlinearity and presence of parameter δ, it is very difficult to solve these problems. Here we solve these classes of important equations using the Chebyshev spectral collocation method. We have provided the convergence of the proposed approximate method. The trueness of the method is shown by applying it to some illustrative examples. Results are compared with some known methods to highlight its neglectable error and high accuracy.

A New Numerical Technique for Solving Volterra Integral Equations Using Chebyshev Spectral Method

In this work, we propose a new method for solving Volterra integral equations. The technique is based on the Chebyshev spectral collocation method. The application of the proposed method leads Volterra integral equation to a system of algebraic equations that are easy to solve. Some examples are presented and compared with some methods in the literature to illustrate the ability of this technique. The results demonstrate that the new method is more efficient, convergent, and accurate to the exact solution.

Instability of a binary liquid film flowing down a slippery inclined heated plate

In this thesis we studied the stability of a binary liquid film flowing down a heated porous inclined plate. It is assumed that the heating induces concentration differences in the liquid mixture (Soret effect), which together with the differences in temperature affects the surface tension. A mathematical model is constructed by coupling the Navier- tokes equations governing the flow with equations for the concentration and temperature. The effect of substrate permeability is incorporated by applying a specific slip condition at the bottom of the liquid layer. We carry out a linear stability analysis in order to obtain the critical conditions for the onset of instability. We used a Chebyshev spectral collocation method to obtain numerical solutions to the resulting Orr-Sommerfeld type equations. We also obtained an asymptotic solution which yielded an expression for the state of neutral stability of long perturbations as a function of the parameters controlling the problem. We present our findings by illustrating and interpreting our results for the critical Reynolds number for instability.

Instability of a binary liquid film flowing down a slippery inclined heated plate

In this thesis we studied the stability of a binary liquid film flowing down a heated porous inclined plate. It is assumed that the heating induces concentration differences in the liquid mixture (Soret effect), which together with the differences in temperature affects the surface tension. A mathematical model is constructed by coupling the Navier- tokes equations governing the flow with equations for the concentration and temperature. The effect of substrate permeability is incorporated by applying a specific slip condition at the bottom of the liquid layer. We carry out a linear stability analysis in order to obtain the critical conditions for the onset of instability. We used a Chebyshev spectral collocation method to obtain numerical solutions to the resulting Orr-Sommerfeld type equations. We also obtained an asymptotic solution which yielded an expression for the state of neutral stability of long perturbations as a function of the parameters controlling the problem. We present our findings by illustrating and interpreting our results for the critical Reynolds number for instability.

Robust synchronization of chaotic fractional-order systems with shifted Chebyshev spectral collocation method

In this work, synchronization of fractional dynamics of chaotic system is presented. The suggested dynamics is governed by a system of fractional differential equations, where the fractional derivative operator is modeled by the novel Caputo operator. The nature of fractional dynamical system is non-local which often rules out a closed-form solution. As a result, an efficient numerical method based on shifted Chebychev spectral collocation method is proposed. The error and convergence analysis of this scheme is also given. Numerical results are given for different values of fractional order and other parameters when applied to solve chaotic system, to address any points or queries that may occur naturally.