Shelly Arora

Solution of two point boundary value problems using orthogonal collocation on finite elements

A convenient computational algorithm to solve boundary value problems (BVP) using orthogonal collocation on finite elements (OCFE) is presented. The algorithm is the conjunction of finite element method (FEM) and orthogonal collocation method (OCM). OCM discretize BVPs conveniently where as FEM provides accuracy to the solution. The method is applied to the boundary value problems having mixed Robbins and Dirichlets boundary conditions. The numerical results are compared with the analytic solution.

Simulation of washing of packed bed of porous particles by orthogonal collocation on finite elements

Abstract A mathematical model related to diffusion–dispersion during flow through multiparticle system is presented. The technique of orthogonal collocation on finite elements is applied on the axial and radial domain of the system of governing partial differential equations. The convergence and stability of the solutions is also checked. Effect of different parameters like axial dispersion coefficient, reaction rate kinetics, interstitial velocity, bed porosity, cake thickness, particle radius on exit solute concentration is presented. Numerical results are matching the experimental results significantly. The technique is simple, elegant and convenient for solving two point boundary value problems with no limit on the range of parameters.

Application of orthogonal collocation on finite elements for solving non-linear boundary value problems

Abstract A simple, convenient and easy approach to solve non-linear boundary value problems (BVP) using orthogonal collocation on finite elements (OCFE) is presented. The algorithm is the conjunction of finite element method (FEM) and orthogonal collocation method (OCM). The stability of the numerical results is checked by a novel algorithm which not only justifies the stability of the results but also checks the convergence of the method. The method is applied to the non-symmetric boundary value problems having Dirichlet’s and mixed Robbin’s boundary conditions.

A computationally efficient technique for solving two point boundary value problems in porous media

A numerical technique for solving two point boundary value problems in the porous media is presented. The technique is the combination of orthogonal collocation and finite element methods. One provides the accuracy whereas the other provides the stability to the numerical results. The technique is much simpler and the computer codes can be easily developed. The numerical results are compared with the analytic results for verification. The numerical results are found to match the analytic ones for more than four decimal places. The method can be applied to the stiff problems without any limit on the range of parameters.

Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation

The solution and analysis of Kuramoto–Sivashinsky equation by cubic Hermite collocation method is performed and a bound for maximum norm of the semi-discrete solution is derived by using Lyapunov functional. Error estimates are also obtained for semi-discrete solutions and verified by numerical experiments.

Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions

Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.

Applications of Quintic Hermite collocation with time discretization to singularly perturbed problems

Singular perturbation problems have been discussed using collocation technique with quintic Hermite interpolating polynomials as base functions. These polynomials have the property to interpolate the function as well as its tangent at node points. To discretize the problem in temporal direction forward difference operator has been applied. The given technique is a combination of collocation and difference scheme. Parameter uniform convergence has been studied using the method given by Farrell and Hegarty (1991). Rate of convergence of quintic Hermite difference scheme has been found to depend upon node points. Applicability and computational effect of the scheme has been examined through numerical examples. Results have been presented graphically through surface plots as well as in tabular form.

Modelling and Simulation of a Packed Bed of Pulp Fibers Using Mixed Collocation Method

A convenient computational approach for solving mathematical model related to diffusion dispersion during flow through packed bed is presented. The algorithm is based on the mixed collocation method. The method is particularly useful for solving stiff system arising in chemical and process engineering. The convergence of the method is found to be of order 2 using the roots of shifted Chebyshev polynomial. Model is verified using the literature data. This method has provided a convenient check on the accuracy of the results for wide range of parameters, namely, Peclet numbers. Breakthrough curves are plotted to check the effect of Peclet number on average and exit solute concentrations.

Computationally efficient technique for weight functions and effect of orthogonal polynomials on the average

The properties of orthogonal collocation on finite elements with respect to the choice of orthogonal polynomials are studied. A simplified algorithm for the calculation of the collocation points, weight functions and discretization matrices for first and second order derivatives is presented in terms of the Lagrangian interpolation polynomial. The effect of Legendre and Chebyshev polynomials on the average value of the dependent variable is checked. It is found that the Legendre polynomial gives the better results at the centre and on the average as compare to the Chebyshev polynomial.

Solution of Time-Dependent Linear Singular Perturbation Problems Using Collocation Techniques with Hermite Basis

Time-dependent singular perturbation problems having steep gradients near boundary are solved numerically, using the Hermite collocation method. Cubic Hermite polynomials have been taken as interpolating polynomials with two collocation points within each mesh [xi−1, xi] where, i = 1, 2,…, n. Parameter uniform convergence has been carried out using the technique given by Farrell and Hegarty (Farrell, P.A., Hegarty, A., 1991. On the Determination of the Order of Uniform Convergence. In Proceedings of the13th World Congress on Computation and Applied Mathematics, 2, pp. B1-B2). Hermite collocation is applied on different singular perturbation problems. Numerical results are presented in terms of 2D and 3D graphs to analyse the problems. The rate of convergence of Hermite collocation method is found to be dependent on the mesh points. The point wise error and rate of convergence are presented in tabular form.