Mohamed Suleiman

Diagonally implicit block backward differentiation formulas for solving fuzzy differential equations

In this work, the diagonally implicit 2-points block backward differentiation formulas (DIBBDF) is developed for solving Fuzzy Differential Equations (FDEs) under the interpretation of generalized Hukuhara differentiability. The fuzzy configuration of this method is also introduced. Numerical results using DIBBDF are presented and compared with the existing method. It is clearly shown that the proposed method obtains good numerical results and suitable for solving FDEs.

Numerical solution for solving second order ordinary differential equations using block method

The purpose of this paper is to present a four point direct block one-step method for solving directly the general second order nonstiff initial value problems (IVPs) of ordinary differential equations (ODEs). The mathematical problems in real world can be written in the form of differential equations and arise in the fields of science and engineering such as fluid dynamic, electric circuit, motion of rocket or satellite and other area of application. The proposed method will estimate the approximation solutions at four points simultaneously by using variable step size. Numerical results are given to show the efficiency of the proposed method.

An embedded 5(4) explicit Runge-Kutta-Nyström method with dissipation of high order

A new 5(4) embedded explicit Runge-Kutta-Nystrom (RKN) method with dissipation of high order is developed to solve integration of initial-value problems for second-order ordinary differential equations possessing oscillating solutions. The fifth order formula has dispersive order eight and dissipative order nine. The fourth-order embedded formula is obtained to control the local truncation errors. Numerical experiments indicate that the new method is more efficient than the existing embedded explicit RKN methods.

ON THE SOLUTION OF TWO POINT BOUNDARY VALUE PROBLEMS WITH TWO POINT DIRECT METHOD

The two point boundary value problems (BVPs) occur in a wide variety of applications especially in sciences such as chemistry and biology. In this paper, we propose two point direct method of order six for solving nonlinear two point boundary value problems directly. This method is presented in a simple form of Adams Mouton type and determines the approximate solution at two point simultaneously. The method will be implemented using constant step size via shooting technique adapted with three-step iterative method. Numerical results are given to compare the efficiency of the proposed method with the Runge-Kutta and bvp4c method.

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.

Order and stability of 2-point block backward difference method

This paper studied the order and stability of a 2-Point Block Backward Difference method (2PBBD) for solving systems of nonstiff higher order Ordinary Differential Equations (ODEs) directly. The me...

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.

Numerical solution of differential algebraic equations (DAEs) by mix-multistep method

Differential Algebraic Equations (DAEs) are regarded as stiff Ordinary Differential Equations (ODEs). Therefore they are solved using implicit method such as Backward Differentiation Formula (BDF) type of methods which require the use of Newton iteration which need much computational effort. However, not all of the ODEs in DAE system are stiff. In this paper, we describe a new technique for solving DAE, where the ODEs are treated as non-stiff at the start of the integration and putting the non-stiff ODEs into stiff subsystem should instability occurs. Adams type of method is used to solve the non-stiff part and BDF method for solving the stiff part. This strategy is shown to be competitive in terms of computational effort and accuracy. Numerical experiments are presented to validate its efficiency.

The arithmetic mean iterative method for solving 2D Helmholtz equation

In this paper, application of the Arithmetic Mean (AM) iterative method is extended by solving second order finite difference algebraic equations. The performance of AM method in solving second order finite difference algebraic equations is comparatively studied by their application on two-dimensional Helmholtz equation. Numerical results of AM method in solving two test problems are included and compared with the standard Gauss-Seidel (GS) method. Based on the numerical results obtained, the results show that AM method is better than GS method in the sense of number of iterations and CPU time.

Solving DAEs using block method

This paper is on solving semi-explicit index-one Differential Algebraic Equations (DAEs). The block method suggested computes the solutions of the DAE at 2-point simultaneously. The numerical results obtained are compared with non-block backward differentiation method (BDF). The comparison of the numerical results confirms that the block method developed is more efficient and accurate.