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A class of hybrid methods for solving fourth-order ordinary differential equations (HMFD) is proposed and investigated. Using the theory of B-series, we study the order of convergence of the HMFD methods. The main result is a set of order conditions, analogous to those for two-step hybrid method, which offers a better alternative to the usual ad hoc Taylor expansions. Based on the algebraic order conditions, a one-stage and two-stage explicit HMFD methods are constructed. Results from numerical experiment suggest the superiority of the new methods in terms of accuracy and computational efficiency over hybrid methods for special second ODEs, Runge-Kutta methods recently proposed for solving special fourth-order ODEs directly and some linear multistep methods proposed for the same purpose in the literature. [ABSTRACT FROM AUTHOR]

This paper proposes and investigates a special class of explicit Runge-Kutta-Nyström (RKN) methods for problems in the form y′′(x)=f(x,y,y′) including third derivatives and denoted as STDRKN. The methods involve one evaluation of second derivative and many evaluations of third derivative per step. In this study, methods with two and three stages of orders four and five, respectively, are presented. The stability property of the methods is discussed. Numerical experiments have clearly shown the accuracy and the efficiency of the new methods. [ABSTRACT FROM AUTHOR]

The problems of multiple inclined or curved cracks in circular positions is treated by using the hypersingular integral equation method. The cracks center are placed at the edge of a virtual circle with radius R. The first crack is fixed on the x-axis while the second crack is located on the boundary of a circle with the varying angle, θ. A system of hypersingular integral equations is formulated and solved numerically for the stress intensity factor (SIF). Numerical examples demonstrate the effect of interaction between two cracks in circular positions are given. It is found that, the severity at the second crack tips are significant when the ratio length of the second to the first crack is small and it is placed at a small angle of θ. [ABSTRACT FROM AUTHOR]

In this paper, an improved explicit two-step hybrid method with fifth algebraic order is derived. The new method possesses dispersion of order 10 and dissipation of order seven, which is first of its kind in the literature. Numerical experiment reveals the superiority of the new method for solving oscillatory or periodic problems over several methods of the same algebraic order. [ABSTRACT FROM AUTHOR]

Two-step optimized hybrid methods of order five and order six are developed for the integration of second order oscillatory initial value problems. The optimized hybrid method (OHMs) are based on the existing nonzero dissipative hybrid methods. Phase-lag, dissipation or amplification error, and the differentiation of the phase-lag relations are required to obtain the methods. Phase-fitted methods based on the same nonzero dissipative hybrid methods are also constructed. Numerical results show that OHMs are more accurate compared to the phase-fitted methods and some well-known methods appeared in the scientific literature in solving oscillating second order initial value problems. It is also found that the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper three pairs of embedded Runge–Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. The second method is the RKD5(4) pair followed by the RKD6(5) pair. The methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates. Variables stepsize codes are developed based on the methods and used to solve a set of special third order problems. Numerical results are compared with the existing embedded Runge–Kutta pairs which require the problems to be reduced into a system of first order ODEs. Numerical results have clearly shown the advantage and the efficiency of the new RKD pairs. [Copyright &y& Elsevier]

Runge-Kutta-Nyström (RKN) method is adapted for solving the special second order delay differential equations (DDEs). The stability polynomial is obtained when this method is used for solving linear second order delay differential equation. A standard set of test problems is solved using the method together with a cubic interpolation for evaluating the delay terms. The same set of problems is reduced to a system of first order delay differential equations and then solved using the existing Runge-Kutta (RK) method. Numerical results show that the RKN method is more efficient in terms of accuracy and computational time when compared to RK method. The methods are applied to a well-known problem involving delay differential equations, that is, the Mathieu problem. The numerical comparison shows that both methods are in a good agreement. [ABSTRACT FROM AUTHOR]

We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and Runge- Kutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented. [ABSTRACT FROM AUTHOR]

New implicit block formulae that compute solution of stiff initial value problems at two points simultaneously are derived and implemented in a variable step size mode. The strategy for changing the step size for optimum performance involves halving, increasing by a multiple of 1.7, or maintaining the current step size. The stability analysis of the methods indicates their suitability for solving stiff problems. Numerical results are given and compared with some existing backward differentiation formula algorithms. The results indicate an improvement in terms of accuracy. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]