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.

Vibration characteristics of multiple functionally graded nonuniform beams

Based on the Euler–Bernoulli beam theory, the natural vibration behavior of functionally graded nonuniform multiple beams has been investigated. It is assumed that the beams are joined by elastic translational springs, and the properties of the beams vary along the axial direction. The Chebyshev spectral collocation method has been used to convert the governing differential equations of transverse motion into a system of algebraic equations that are put in the matrix–vector form. Then, the dimensionless transverse frequencies are obtained by solving the eigenvalue problem. The influence of several factors, such as the stiffness parameters of the coupling translational springs, the properties of the cross section of the beams, and the boundary conditions on the frequencies, has been carried out. The results generated from the Chebyshev spectral collocation method have been verified by comparing them with those reported in other studies and references from the literature. Several numerical examples have been presented and discussed to analyze the system under consideration. The authors hope that the findings of the current study are helpful in designing and characterizing multiple nonuniform thin engineering structures.

Electromagnetohydrodynamic Electroosmotic Flow and Entropy Generation of Third-Grade Fluids in a Parallel Microchannel

The present paper discusses the electromagnetohydrodynamic (EMHD) electroosmotic flow (EOF) and entropy generation of incompressible third-grade fluids in a parallel microchannel. Numerical solutions of the non-homogeneous partial differential equations of velocity and temperature are obtained by the Chebyshev spectral collocation method. The effects of non-Newtonian parameter Λ, Hartman number Ha and Brinkman number Br on the velocity, temperature, Nusselt number and entropy generation are analyzed in detail and shown graphically. The main results show that both temperature and Nusselt number decrease with the non-Newtonian physical parameter, while the local and total entropy generation rates exhibit an adverse trend, which means that non-Newtonian parameter can provoke the local entropy generation rate. In addition, we also find that the increase of non-Newtonian parameter can lead to the increase of the critical Hartman number Hac.