Khairil Iskandar Othman

A quantitative comparison of numerical method for solving stiff ordinary differential equations

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLABs suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

A numerical algorithm for solving stiff ordinary differential equations

An advanced method using block backward differentiation formula (BBDF) is introduced with efficient strategy in choosing the step size and order of the method. Variable step and variable order block backward differentiation formula (VSVO-BBDF) approach is applied throughout the numerical computation. The stability regions of the VSVO-BBDF method are investigated and presented in distinct graphs. The improved performances in terms of accuracy and computation time are presented in the numerical results with different sets of test problems. Comparisons are made between the proposed method and MATLAB’s suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

Derivation of diagonally implicit block backward differentiation formulas for solving stiff initial value problems

The diagonally implicit 2-point block backward differentiation formulas (DI2BBDF) of order two, order three, and order four are derived for solving stiff initial value problems (IVPs). The stability properties of the derived methods are investigated. The implementation of the method using Newton iteration is also discussed. The performance of the proposed methods in terms of maximum error and computational time is compared with the fully implicit block backward differentiation formulas (FIBBDF) and fully implicit block extended backward differentiation formulas (FIBEBDF). The numerical results show that the proposed method outperformed both existing methods.

Numerical solution of second order ODE directly by two point block backward differentiation formula

Direct Two Point Block Backward Differentiation Formula, (BBDF2) for solving second order ordinary differential equations (ODEs) will be presented throughout this paper. The method is derived by differentiating the interpolating polynomial using three back values. In BBDF2, two approximate solutions are produced simultaneously at each step of integration. The method derived is implemented by using fixed step size and the numerical results that follow demonstrate the advantage of the direct method as compared to the reduction method.

Stability region of 3-point block backward differentiation formula

In this paper, we focus on the stability region of the variable step size of 3-point block backward differentiation formula (VSBBDF) method. The graphs are plotted using MAPLE software. To show the performance of the method, the accuracy is presented in the numerical results to solve first order stiff ordinary differential equations (ODEs).

Numerical solution of a linear Klein-Gordon equation

A new scheme of a linear inhomogeneous Klein-Gordon equation is developed by utilizing finite difference method incorporated with arithmetic mean averaging of functional values. This study considered the central time central space (CTCS) finite difference scheme incorporated with four points arithmetic mean averaging. In addition, the theoretical aspects of finite difference scheme are also considered such as stability, consistency and convergence. The von Neumann stability analysis method and Miller Norm Lemma are used to analyze the stability of the proposed scheme. The performance analysis shows the proposed scheme is stable, consistent and convergent. These theoretical analyses are verified by a numerical experiment. The comparison results shown the proposed scheme produces better accuracy rather than the standard CTCS scheme.

NUMERICAL SOLUTION OF TUMOR-IMMUNE INTERACTION USING 2-POINT BLOCK BACKWARD DIFFERENTIATION METHOD

In this paper, we consider tumor-immune interaction model systems. The numerical solutions for the tumor-immune interaction system are obtained by using the 2-point Block Backward Differentiation Formula (BBDF) methods developed by Zarina et al. in 2007. The numerical results are presented in terms of computational time and accuracy of the solutions.

Non-convergence partitioning strategy for solving van der Pol’s equations

Partitioning is a strategy that will reduce computational cost. Starting with all equations be treated as non-stiff, this strategy will divide the equations into the stiff and nonstiff subsystem. The non-convergence partitioning strategy will determine equations that caused instability, and put the equations into stiff subsystem and solved using Newton iteration backward differentiation formulae and all other equations remain in the non-stiff subsystem and solved by Adams method. This partitioning strategy will continue until instability occurs again and placed equations that caused instability into the stiff subsystem. But for van der Pol equation, the nature of the equations need to change from stiff to the non-stiff subsystem and vice versa when it is necessary. This paper will extend the non-convergence partitioning strategy by allowing equations from the stiff subsystem to be placed back in the non-stiff subsystem.

Numerical solution of linear Klein-Gordon equation using FDAM scheme

Many scientific areas appear in a hyperbolic partial differential equation like the Klien-Gordon equation. The analytical solutions of the Klein-Gordon equation have been approximated by the suggested numerical approaches. However, the arithmetic mean (AM) method has not been studied on the Klein-Gordon equation. In this study, a new proposed scheme has utilized central finite difference formula in time and space (CTCS) incorporated with AM formula averaging of functional values for approximating the solutions of the Klein-Gordon equation. Three-point AM is considered to a linear inhomogeneous Klein-Gordon equation. The theoretical aspects of the numerical scheme for the Klein-Gordon equation are also considered. The stability analysis is analyzed by using von Neumann stability analysis and Miller Norm Lemma. Graphical results verify the necessary conditions of Miller Norm Lemma. Good results obtained relate to the theoretical aspects of the numerical scheme. The numerical experiments are examined to verify the theoretical analysis. Comparative study shows the new CTCS scheme incorporated with three-point AM method produced better accuracy and shown its reliable and efficient over the standard CTCS scheme.Many scientific areas appear in a hyperbolic partial differential equation like the Klien-Gordon equation. The analytical solutions of the Klein-Gordon equation have been approximated by the suggested numerical approaches. However, the arithmetic mean (AM) method has not been studied on the Klein-Gordon equation. In this study, a new proposed scheme has utilized central finite difference formula in time and space (CTCS) incorporated with AM formula averaging of functional values for approximating the solutions of the Klein-Gordon equation. Three-point AM is considered to a linear inhomogeneous Klein-Gordon equation. The theoretical aspects of the numerical scheme for the Klein-Gordon equation are also considered. The stability analysis is analyzed by using von Neumann stability analysis and Miller Norm Lemma. Graphical results verify the necessary conditions of Miller Norm Lemma. Good results obtained relate to the theoretical aspects of the numerical scheme. The numerical experiments are examined to veri...

Intervalwise block partitioning for 3-point in solving linear systems of first order ordinary differential equations

Intervalwise partitioning is a strategy to solve stiff ordinary differential equations (ODEs). This strategy using on 3-point block method will initially starts solving ODE using Adams method, and switch the system to Backward Differentiation Formula (BDF) when there is an indication of stiffness. Indication of stiffness will be based on hacc > hiter and the trace of the Jacobian. The comparison with existing method reveals that this partitioning strategy can be an alternative method to solve stiff ODEs.