Fourth order compact finite difference method for solving singularly perturbed 1D reaction diffusion equations with dirichlet boundary conditions

A numerical method based on finite difference scheme with uniform mesh is presented for solving singularly perturbed two-point boundary value problems of 1D reaction-diffusion equations. First, the derivatives of the given differential equation is replaced by the finite difference approximations and then, solved by using fourth order compact finite difference method by taking uniform mesh. To demonstrate the efficiency of the method, numerical illustrations have been given. Graphs are also depicted in support of the numerical results. Both the theoretical and computational rate of convergence of the method have been examined and found to be in agreement. As it can be observed from the numerical results presented in tables and graphs, the present method approximates the exact solution very well.Keywords: Singular perturbation, Compact finite difference method, Reaction diffusion

Tchebychev Polynomial Approximations for mthm^{th} Order Boundary Value Problems

Higher order boundary value problems (BVPs) play an important role modeling various scientific and engineering problems. In this article we develop an efficient numerical scheme for linear $m^{th}$ order BVPs. First we convert the higher order BVP to a first order BVP. Then we use Tchebychev orthogonal polynomials to approximate the solution of the BVP as a weighted sum of polynomials. We collocate at Tchebychev clustered grid points to generate a system of equations to approximate the weights for the polynomials. The excellency of the numerical scheme is illustrated through some examples.Comment: 21 pages, 10 figure

Static Response and Buckling Loads of Multilayered Composite Beams Using the Refined Zigzag Theory and Higher-Order Haar Wavelet Method

The paper presents a review of Haar wavelet methods and an application of the higher-order Haar wavelet method to study the behavior of multilayered composite beams under static and buckling loads. The Refined Zigzag Theory (RZT) is used to formulate the corresponding governing differential equations (equilibrium/stability equations and boundary conditions). To solve these equations numerically, the recently developed Higher-Order Haar Wavelet Method (HOHWM) is used. The results found are compared with those obtained by the widely used Haar Wavelet Method (HWM) and the Generalized Differential Quadrature Method (GDQM). The relative numerical performances of these numerical methods are assessed and validated by comparing them with exact analytical solutions. Furthermore, a detailed convergence study is conducted to analyze the convergence characteristics (absolute errors and the order of convergence) of the method presented. It is concluded that the HOHWM, when applied to RZT beam equilibrium equations in static and linear buckling problems, is capable of predicting, with a good accuracy, the unknown kinematic variables and their derivatives. The HOHWM is also found to be computationally competitive with the other numerical methods considered

Free longitudinal vibrations of functionally graded tapered axial bars by pseudospectral method

In this work, the problem of free longitudinal vibration of rods with variable cross-sectional area and material properties is investigated using the pseudospectral method. With the gradation of material properties like modulus of elasticity and mass density in the axial direction, the results corresponding to a functionally graded axial bar are obtained using the proposed pseudospectral formulation. The pseudospectral formulation used is relatively easy to implement and powerful in analyzing vibration problems. With the help of several numerical examples, the non-dimensional natural frequencies of rods obtained using the pseudospectral method are compared with those obtained by the analytical solution, generalized finite element method, the discrete singular convolution method and differential transformation method. The numerical results obtained show that the proposed technique allows boundary conditions to be incorporated easily and yields results with good accuracy and faster convergence rates than other methods

Mechanical Engineering

The book substantially offers the latest progresses about the important topics of the "Mechanical Engineering" to readers. It includes twenty-eight excellent studies prepared using state-of-art methodologies by professional researchers from different countries. The sections in the book comprise of the following titles: power transmission system, manufacturing processes and system analysis, thermo-fluid systems, simulations and computer applications, and new approaches in mechanical engineering education and organization systems

On the use of exponential basis functions in the analysis of shear deformable laminated plates

In this report, we introduce a meshfree approach for static analysis of isotropic/orthotropic crossply laminated plates with symmetric/non-symmetric layers. Classical, first and third order shear deformation plate theories are employed to perform the analyses. In this method, the solution is first split into homogenous and particular parts and then the homogenous part is approximated by the summation of an appropriately selected set of exponential basis functions (EBFs) with unknown coefficients. In the homogenous solution the EBFs are restricted to satisfy the governing differential equation. The particular solution is derived using a similar approach and another series of EBFs. The imposition of the boundary conditions and determination of the unknown coefficients are performed by a collocation method through a discrete transformation technique. The solution method allows us to obtain semi-analytical solution of plate problems with various shapes and boundary conditions. The solutions of several benchmark plate problems with various geometries are presented to validate the results

A Modified Spectral Relaxation Method for Some Emden-Fowler Equations

In this chapter, we present a modified version of the spectral relaxation method for solving singular initial value problems for some Emden-Fowler equations. This study was motivated by the several applications that these equations have in Science. The first step of the method of solution makes use of linearisation to solve the model problem on a small subinterval of the problem domain. This subinterval contains a singularity at the initial instant. The first step is combined with using the spectral relaxation method to recursively solve the model problem on the rest of the problem domain. We make use of examples to demonstrate that the method is reliable, accurate and computationally efficient. The numerical solutions that are obtained in this chapter are in good agreement with other solutions in the literature