V. K. Kukreja

On a class of quadratically convergent iteration formulae

In this paper a new class of iterative formulae having quadratic convergence is presented. Furthermore, these algorithms are comparable to the well known method of Newton and the computed results support this theory.

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.

On some third-order iterative methods for solving nonlinear equations

Using the iteration formulas of second order [Mamta, V. Kanwar, V.K. Kukreja, Sukhjit Singh, On a class of quadratically convergent iteration formulae, Appl. Math. Comput. 2004, in press] for solving single variable nonlinear equations, two classes of third-order multipoint methods without using second derivative are derived. The main advantage of these classes is that they do not fail if the derivative of the function is either zero or very small in the vicinity of the required root. Further, a new family of Secant-like method with guaranteed super linear convergence is obtained by discrete modifications and their comparison with respect to the existing classical methods is given.

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.

Numerical solution of Burgers’ equation by cubic Hermite collocation method

Abstract In this paper, numerical solution of the non-linear Burgers’ equation are obtained by using cubic Hermite collocation method (CHCM). The advantage of the method is continuity of the dependent variable and its derivative throughout the solution range. A linear stability analysis shows that the numerical scheme based on Crank–Nicolson approximation in time is unconditionally stable. This method is applied on some test problems, with different choice of collocation points to validate the accuracy of the method. The obtained numerical results show that the method is efficient, robust and reliable even for high Reynolds numbers, for which the exact solution fails. Moreover, the method can be applied to a wide class of nonlinear partial differential equations.

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.

Numerical approach for solving diffusion problems using cubic B-spline collocation method

Abstract A numerical approach for modeling the removal of homogeneous solutes from the porous structure is presented. Pore diffusion equations, controlling the overall process, are expressed in terms of concentration of solute as a function of position in bed and washing period. A computational scheme, specifying knot vector(s) and control points (de Boor points), to solve diffusion model using cubic B-spline collocation method (CSCM) is given. The knot vector provides sufficient resolution and control points yield an accurate approximation of the desired curve. Convergence of numerical scheme is of O ( h 2 ) . Four problems have been worked out by the proposed scheme. Results are plotted graphically for fairly wide range of parameters. A comparison with the previous techniques demonstrates the superiority of the proposed method on basis of computational time, efficiency and accuracy.

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.