Fudziah Ismail

Embedded diagonally implicit Runge-Kutta-Nystrom 4(3) pair for solving special second-order IVPs

Abstract In this paper, third-order 3-stage diagonally implicit Runge–Kutta–Nystrom method embedded in fourth-order 4-stage for solving special second-order initial value problems is constructed. The method has the property of minimized local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The stability of the method is investigated and a standard set of test problems are tested upon and comparisons on the numerical results are made when the same set of test problems are reduced to first-order system and solved using existing Runge–Kutta method. The results clearly shown the advantage and the efficiency of the new method.

An Eigenvalue-Eigenvector Method for Solving a System of Fractional Differential Equations with Uncertainty

A new method is proposed for solving systems of fuzzy fractional differential equations (SFFDEs) with fuzzy initial conditions involving fuzzy Caputo differentiability. For this purpose, three cases are introduced based on the eigenvalue-eigenvector approach; then it is shown that the solution of system of fuzzy fractional differential equations is vector of fuzzy-valued functions. Then the method is validated by solving several examples.

A Three-Stage Fifth-Order Runge-Kutta Method for Directly Solving Special Third-Order Differential Equation with Application to Thin Film Flow Problem.

In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.

Third-Order Improved Runge-Kutta Method for Solving Ordinary Differential Equation

this paper we constructed the sets of explicit third-order Improved Runge-Kutta (IRK) methods. The method used in two and three stage which indicated as the required number of function evaluations per step. The third-order IRK method in two-stage has a lower number of function evaluations than the classical third-order RK method while maintaining the same order of local accuracy. In three-stages, the new method is more accurate compared to the classical third-order RK method. The stability region of methods are given and numerical examples are presented to illustrate the efficiency and accuracy of the new methods.

Solving delay differential equations using componentwise partitioning by Runge-Kutta method

Embedded singly diagonally implicit Runge-Kutta (SDIRK) method is used to solve stiff systems of delay differential equations (DDEs). The delay argument is approximated using Hermite interpolation. Initially the whole system is considered as nonstiff and solved using simple iteration. When stiffness is indicated, the appropriate equation is placed into the stiff subsystem and solved using Newton iteration. This type of partitioning is called componentwise partitioning. The process is continued until all the equations have been placed in the right subsystem. Numerical results based on componentwise partitioning and intervalwise partitioning are tabulated and compared.

Zero-dissipative phase-fitted hybrid methods for solving oscillatory second order ordinary differential equations

In this paper, zero-dissipative phase-fitted two-step hybrid methods are developed for the integration of second-order periodic initial value problems. The phase-fitted hybrid methods are constructed using similar approaches introduced by Papadopoulos et al. [1]. This new methods are based on the existing explicit hybrid methods of order four and six. Numerical illustrations indicate that the new methods are much more efficient than the existing methods.

Mixed Convection Boundary Layer Flow Embedded in a Thermally Stratified Porous Medium Saturated by a Nanofluid

We present the numerical investigation of the steady mixed convection boundary layer flow over a vertical surface embedded in a thermally stratified porous medium saturated by a nanofluid. The governing partial differential equations are reduced to the ordinary differential equations, using the similarity transformations. The similarity equations are solved numerically for three types of metallic or nonmetallic nanoparticles, namely, copper (Cu), alumina (Al2O3), and titania (TiO2), in a water-based fluid to investigate the effect of the solid volume fraction or nanoparticle volume fraction parameter φ of the nanofluid on the flow and heat transfer characteristics. The skin friction coefficient and the velocity and temperature profiles are presented and discussed.

Radiation effect on Marangoni convection boundary layer flow of a nanofluid

PurposeIn this paper, we present a mathematical model for Marangoni convection boundary layer flow with radiation and different types of nanoparticles, namely, Cu, Al2O3, and TiO2 in a water-based fluid.MethodThe governing equations in the form of partial differential equations have been reduced to a set of ordinary differential equations by applying suitable similarity transformations, which is then solved numerically using the shooting method.ResultsNumerical results are obtained for the surface-temperature gradient or the heat transfer rate as well as the temperature profiles for some values of the governing parameters, namely, the nanoparticle volume fraction φ, the constant exponent β, and thermal radiation parameter Nr.ConclusionThe results indicate that the heat transfer rate at the surface decreases as the thermal radiation parameter Nr increases.MSC: 76N20, fluid mechanics.

Numerical Solution of Second-Order Fuzzy Differential Equation Using Improved Runge-Kutta Nystrom Method

We develop the Fuzzy Improved Runge-Kutta Nystrom (FIRKN) method for solving second-order fuzzy differential equations (FDEs) based on the generalized concept of higher-order fuzzy differentiability. The scheme is two-step in nature and requires less number of stages which leads to less number of function evaluations in comparison with the existing Fuzzy Runge-Kutta Nystrom method. Therefore, the new method has a lower computational cost which effects the time consumption. We assume that the fuzzy function and its derivative are Hukuhara differentiable. FIRKN methods of orders three, four, and five are derived with two, three, and four stages, respectively. The numerical examples are given to illustrate the efficiency of the methods.

A new variable step size block backward differentiation formula for solving stiff initial value problems

A new block backward differentiation formula of order 4 with variable step size is formulated. By varying a parameter in the formula, different sets of formulae with A-stability property can be generated. At the cost of an additional function evaluation, the accuracy of the method is seen to outperform some existing backward differentiation formula algorithms. The strategy involved in controlling the step size ratio is also described. The problems tested with the method show its efficiency in solving stiff initial value problems.