2 point block backward difference method for solving Riccati type differential problems

A two point block backward difference method is established to solve Riccati differential equations directly. Based on a predictor-corrector two point block backward difference method (2PBBD), a code is developed using a set of integration coefficients that eliminates the need to be calculated at every step change. The method requires calculating the integration coefficients only once in the beginning. The 2PBBD has an added advantage of a recurrence relationship between coefficients of different orders which provides a more elegant algorithm. The recurrence relationship between coefficients also reduces the computational cost

Solving Nonstiff Higher-Order Ordinary Differential Equations Using 2-Point Block Method Directly

We describe the development of a 2-point block backward difference method (2PBBD) for solving system of nonstiff higher-order ordinary differential equations (ODEs) directly. The method computes the approximate solutions at two points simultaneously within an equidistant block. The integration coefficients that are used in the method are obtained only once at the start of the integration. Numerical results are presented to compare the performances of the method developed with 1-point backward difference method (1PBD) and 2-point block divided difference method (2PBDD). The result indicated that, for finer step sizes, this method performs better than the other two methods, that is, 1PBD and 2PBDD

Adaptive step size of diagonally implicit block backward differentiation formulas for solving first and second order stiff ordinary differential equations with applications

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Direct two-point block methods for solving nonstiff higher order Ordinary Differential Equations using Backward Difference formulation

This thesis describes the development of a Two-Point Block Backward Difference method (2PBBD) for solving system of nonstiff higher order Ordinary Differential Equations (ODEs) directly. The method computes the approximate solutions; and at two points and simultaneously within an equidistant block. This method has the advantages of calculating the integration coefficients only once at the beginning of the integration. The relationship between the explicit and implicit coefficients has also been derived. These motivate us to formulate the association between the formula for predictor and corrector. The relationship between the lower and higher order derivative also have been established. New explicit and implicit block methods using constant step sizes and three back values have also been derived. The algorithm developed is implemented using Microsoft Visual C++ 6.0 and run by High Performance Computer (HPC) using the Message Passing Interface (MPI) library. The stability properties for the 2PBBD methods are analyzed to ensure its suitability for solving nonstiff Initial Value Problems (IVPs). The stability analysis shows that the method is stable.Numerical results are presented to compare the performances of this method with the previously published One-Point Backward Difference (1PBD) and Two-Point Block Divided Difference (2PBDD) methods. The numerical results indicated that for finer step sizes, 2PBBD performs better than 1PBD and 2PBDD

Variable order variable stepsize algorithm for solving nonlinear Duffing oscillator

Nonlinear phenomena in science and engineering such as a periodically forced oscillator with nonlinear elasticity are often modeled by the Duffing oscillator (Duffing equation). The Duffling oscillator is a type of nonlinear higher order differential equation. In this research, a numerical approximation for solving the Duffing oscillator directly is introduced using a variable order stepsize (VOS) algorithm coupled with a backward difference formulation. By selecting the appropriate restrictions, the VOS algorithm provides a cost efficient computational code without affecting its accuracy. Numerical results have demonstrated the advantages of a variable order stepsize algorithm over conventional methods in terms of total steps and accuracy

Variable order variable stepsize algorithm for solving nonlinear Duffing oscillator

Nonlinear phenomena in science and engineering such as a periodically forced oscillator with nonlinear elasticity are often modeled by the Duffing oscillator (Duffing equation). The Duffling oscillator is a type of nonlinear higher order differential equation. In this research, a numerical approximation for solving the Duffing oscillator directly is introduced using a variable order stepsize (VOS) algorithm coupled with a backward difference formulation. By selecting the appropriate restrictions, the VOS algorithm provides a cost efficient computational code without affecting its accuracy. Numerical results have demonstrated the advantages of a variable order stepsize algorithm over conventional methods in terms of total steps and accuracy