Faranak Rabiei

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.

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 novel design of 5-input majority gate in quantum-dot cellular automata technology

Quantum-dot Cellular Automata (QCA) technology is one of the most important technologies, which can be suitable replacement for conventional technologies at Nano-scale. The principle logic elements in the QCA technology are majority gates and inverters. In this paper, a novel design is proposed for 5-input majority gate in the QCA technology. The proposed 5-input majority gate uses half distance. The QCADesigner tool version 2.0.3 is utilized for verifying functionality and layout of the proposed majority gate. The simulation results demonstrate that the proposed 5-input majority gate design provides significant improvements in the logical circuit design in terms of area and the number of required cells in comparison with other majority gates.

Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives

In this paper, the solution of fuzzy differential equations is approximated numerically using diagonally implicit multistep block method of order four. The multistep block method is well known as an efficient and accurate method for solving ordinary differential equations, hence in this paper the method will be used to solve the fuzzy initial value problems where the initial value is a symmetric triangular fuzzy interval. The triangular fuzzy number is not necessarily symmetric, however by imposing symmetry the definition of a triangular fuzzy number can be simplified. The symmetric triangular fuzzy interval is a triangular fuzzy interval that has same left and right width of membership function from the center. Due to this, the parametric form of symmetric triangular fuzzy number is simple and the performing arithmetic operations become easier. In order to interpret the fuzzy problems, Seikkala’s derivative approach is implemented. Characterization theorem is then used to translate the problems into a system of ordinary differential equations. The convergence of the introduced method is also proved. Numerical examples are given to investigate the performance of the proposed method. It is clearly shown in the results that the proposed method is comparable and reliable in solving fuzzy differential equations.

Numerical simulation of fuzzy differential equations using general linear method and B-series:

In this article, numerical simulation of fuzzy differential equations using general linear method is proposed. The significance of general linear method is derivation of algebraic order conditions of method using technique of rooted trees and B-series. Fuzzy general linear method of order 3 based on the concept of generalized Hukuhara differentiability for solving fuzzy differential equations is developed. Convergence of third-order fuzzy general linear method is proven. The proposed method is tested on fuzzy initial value problems, and the numerical results showed that fuzzy general linear method produced more accurate approximation of fuzzy solution for tested problems compared with the existing fuzzy numerical methods.

Composite group of explicit Runge-Kutta methods

In this paper,the composite groups of Runge-Kutta (RK) method are proposed. The composite group of RK method of third and second order, RK3(2) and fourth and third order RK4(3) base on classical Runge-Kutta method are derived. The proposed methods are two-step in nature and have less number of function evaluations compared to the existing Runge-Kutta method. The order conditions up to order four are obtained using rooted trees and composite rule introduced by J. C Butcher. The stability regions of RK3(2) and RK4(3) methods are presented and initial value problems of first order ordinary differential equations are carried out. Numerical results are compared with existing Runge-Kutta method.

Optimized fourth-order Runge-Kutta method for solving oscillatory problems

In this article, we develop a Runge-Kutta method with invalidation of phase lag, phase lag’s derivatives and amplification error to solve second-order initial value problem (IVP) with oscillating solutions. The new method depends on the explicit Runge-Kutta method of algebraic order four. Numerical tests from its implementation to well-known oscillatory problems illustrate the robustness and competence of the new method as compared to the well-known Runge-Kutta methods in the scientific literature.

Fourth order 4-stages improved Runge-Kutta method with minimized error norm

In this paper the improved Runge-Kutta method of order four with 4-stages for solving first order ordinary differential equation is proposed. The method is based on classical Runge-Kutta (RK) method also can be considered as special class of two-step method. Here, the coefficients of the method are obtained using the minimization of the error norm up to order five. The improved method with only 4-stages is more accurate than fourth order 4-stages RK method. Therefore it is computationally more efficient than the existing RK method. A number of test problems are solved and the numerical results compared with the existing RK method are given.

Numerical solution of Fuzzy Differential Equation Using Improved Runge-Kutta Method

In this paper a fuzzy Improved Runge-Kutta method for solving first-order fuzzy differential equations is proposed. The scheme is two step in nature and is based on the Improved Runge-kutta method for solving ordinary differential equations. Here, the fourth order method with three stages is explained. In this method some new parameter be exploited to increase the accuracy in comparison with other same stage existing methods. The convergence of the method is proven, and several numerical examples are experienced to illustrate the effectiveness of the method.

Solving Fuzzy Differential Equation Using Fourth-Order Four-Stage Improved Runge–Kutta Method

In this paper the fuzzy improved Runge–Kutta method of order four for solving first-order fuzzy differential equations is proposed. The scheme is two step in nature and is based on the fourth-order improved Runge–Kutta method for solving first-order ordinary differential equations. The numerical examples are tested to illustrate the efficiency of method.