Tag Archives: collocation

Integrator Block Off – Grid Points Collocation Method For Direct Solution Of Second Order Ordinary Differential Equations Using Chebyshev Polynomials As Basis Function (Published)

The numerical computation of differential equations cannot be overemphasized as it is evident in the literatures. It has been observed that analytical solution of some differential equations are intractable, hence there is need to seek for an alternative solution to such equations. Circumventing this problem resulted into an approximate solution otherwise known as numerical solution. There are so many numerical methods that can be used in solving differential equations which include predictor – corrector method which is linear multistep in nature and not self-starting method. In this presentation the focus is on presenting a self-starting multistep method for direct solution of Second Order Ordinary Differential Equations as against the popular predictor – corrector method which needs additional value for starting point which may alter the accuracy of the method. The method is a mixture of grid and off grid collocation point and often refer to as Block linear multistep method.

Keywords: Block Method., Chebyshev Polynomials, Integrator Off –Grid, Interpolation, Predictor – Corrector, collocation

Integrator Block Off – Grid Points Collocation Method For Direct Solution Of Second Order Ordinary Differential Equations Using Chebyshev Polynomials As Basis Function. (Published)

The numerical computation of differential equations cannot be overemphasized as it is evident in the literatures. It has been observed that analytical solution of some differential equations are intractable, hence there is need to seek for an alternative solution to such equations. Circumventing this problem resulted into an approximate solution otherwise known as numerical solution. There are so many numerical methods that can be used in solving differential equations which include predictor – corrector method which is linear multistep in nature and not self-starting method. In this presentation the focus is on presenting a self-starting multistep method for direct solution of Second Order Ordinary Differential Equations as against the popular predictor – corrector method which needs additional value for starting point which may alter the accuracy of the method. The method is a mixture of grid and off grid collocation point and often refer to as Block linear multistep method.

Keywords: Block Method., Chebyshev Polynomials, Integrator Off –Grid, Interpolation, Predictor – Corrector, collocation

Double Perturbation Collocation Method for Solving Fractional Riccati Differential Equations (Published)

In this work, we proposed a computational technique called the Double Perturbation Collocation Method (DPCM) for the numerical solution of fractional Riccati differential equation. The DPCM requires the addition of a perturbation term to the approximate solution in terms of the shifted Chebyshev polynomials basis function. This function is substituted into a slightly perturbed fractional Riccati equation. The fractional derivative is in the Caputo sense. The resulting equation is simplified and then collocated at some equally spaced points. Thus resulted into system of equations which are then solved by implementing Gaussian elimination method for linear to obtain the unknown constants and for the case of nonlinear, Newton linearization scheme of appropriate orders are used to linearize. The values of the constants obtained are then substituted back into the perturbed approximate solution. Results obtained with DPCM compared favourably well with existing results in literature and the exact solutions where such existed in closed form. Some numerical examples are included to illustrate the accuracy, simplicity and computational cost of the method.

Keywords: Riccati., collocation, double - perturbation, fractional, linearization

Multiple Perturbed Collocation Tau-Method For Solving High Order Linear And Non-Linear Boundary Value Problems (Published)

This paper is concerned with the numerical solution of high order linear and nonlinear boundary value problems of ordinary differential equations by Multiple Perturbed Collocation Tau Method (MPCTM). In applying this method to the class of problems mentioned above, we assumed a perturbed approximate solution in terms of Chebyshev Polynomial basis function which is substituted into the special class of the problem considered. Thus, resulting into n-folds integration. After evaluation of n-fold integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Gaussian elimination method which are then substituted back into the approximate solution. The proposed method is tested on several numerical examples, the approximate solutions is in agreement with the exact solution. The approximate results obtained by the proposed method confirm the convergence of numerical solutions and are compared favorably with the existing methods available in literature

Keywords: Accuracy, Boundary Value Problem, Chebyshev, Errors, Multiple, Perturbed, collocation

Multiple Perturbed Collocation Tau-Method For Solving High Order Linear And Non-Linear Boundary Value Problems (Published)

This paper is concerned with the numerical solution of high order linear and nonlinear boundary value problems of ordinary differential equations by Multiple Perturbed Collocation Tau Method (MPCTM).In applying this method to the class of problems mentioned above, we assumed a perturbed approximate solution in terms of Chebyshev Polynomial basis function which is substituted into the special class of the problem considered. Thus, resulting into n-folds integration. After evaluation of n-fold integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Gaussian elimination method which are then substituted back into the approximate solution.The proposed method is tested on several numerical examples, the approximate solutions is in agreement with the exact solution. The approximate results obtained by the proposed method confirm the convergence of numerical solutions and are compared favorably with the existing methods available in literature.

Keywords: Accuracy, Boundary Value Problem, Chebyshev, Errors, Multiple, Perturbed, collocation

DOUBLE PERTURBATION COLLOCATION METHOD FOR SOLVING FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS (Published)

In this work, we proposed a computational technique called the Double Perturbation Collocation Method (DPCM) for the numerical solution of fractional Riccati differential equation. The DPCM requires the addition of a perturbation term to the approximate solution in terms of the shifted Chebyshev polynomials basis function. This function is substituted into a slightly perturbed fractional Riccati equation. The fractional derivative is in the Caputo sense. The resulting equation is simplified and then collocated at some equally spaced points. Thus resulted into system of equations which are then solved by implementing Gaussian elimination method for linear to obtain the unknown constants and for the case of nonlinear, Newton linearization scheme of appropriate orders are used to linearize. The values of the constants obtained are then substituted back into the perturbed approximate solution. Results obtained with DPCM compared favourably well with existing results in literature and the exact solutions where such existed in closed form. Some numerical examples are included to illustrate the accuracy, simplicity and computational cost of the method.

Keywords: Riccati., collocation, double - perturbation, fractional, linearization