The purpose, search solving Methods for reduce cost of Implementation. Types of Implicit Methods and Their Implementation discussed. The generalization of these methods for stiff differential equations using singly implicit methods. It is seen that number of the calculation operator can e reduce.

In this paper, we study the global truncation error of the linear multistep methods (LMM) in terms of local truncation error of the corresponding Runge-Kutta schemes. The key idea is the representation of LMM with a corresponding Runge-Kutta method. For this, we need to consider the multiple step of a linear multistep method as a single step in the corresponding Runge-Kutta method. Therefore, the global error estimation of a LMM through the Runge-Kutta method will be provided. In this estimation, we do not take into account the effects of roundoff errors. The numerical illustrations show the accuracy and efficiency of the given estimation.

Solving optimal control problems (OCP) with analytical methods has usually been difficult or not cost-effective. Therefore, solv-ing these problems requires numerical methods. There are, of course, many ways to solve these problems. One of the methods available to solve OCP is a forward-backward sweep method (FBSM). In this method, the state variable is solved in a forward and co-state variable by a backward method where an explicit Runge{Kutta method (ERK) is often used to solve differential equations arising from OCP. In this pa-per, instead of the ERK method, three hybrid methods based on ERK method of order 3 and 4 are proposed for the numerical approximation of the OCP. Truncation errors and stability analysis of the presented methods are illustrated. Finally, numerical results of the four opti-mal control problems obtained by new methods, which shows that new methods give us more detailed results, are compared with those of ERK approaches of orders 3 and 4 for solving OCP.

In this paper we try to put different practical aspects of the general linear methods discussed in the papers [1,6,7] under one algorithm to show more details of its implementation. With a proposed initial step size strategy this algorithm shows a better performance in some problems. To illustrate the efficiency of the method we consider some standard test problems and report more useful details of step size and order changes, and number of rejected and accepted steps along with relative global errors.

In this paper, a new interval version of Runge-Kutta methods is proposed for time discretization and solving of optimal control problems (OCPs) in the presence of uncertain parameters. A new technique based on interval arithmetic is introduced to achieve the con dence bounds of the system. The proposed method is based on the new forward representation of Hukuhara interval difference and combining it with Runge-Kutta method for solving the OCPs with interval uncertainties. To perform the proposed method on OCPs, the Lagrange multiplier method is  rst applied to achieve the necessary conditions and then, using some algebraic manipulations, they are converted to an ordinary differential equation to achieve the interval optimal solution for the considered OCP with uncertain parameters. Shooting method is also employed to cover the Runge-Kutta methods restrictions in solving the OCPs with boundary values. The simulation results are applied to some practical case studies for demonstrating the effectiveness of the proposed method.

In this paper, an explicit family with higher-order of accuracy is proposed for dynamic analysis of structural and mechanical systems. By expanding the analytical amplification matrix into Taylor series, the Runge-Kutta family with stages can be presented. The required coefficients ( ) for different stages are calculated through a solution of nonlinear algebraic equations. The contribution of the new family is the equality between its accuracy order, and the number of stages used in a single time step ( ). As a weak point, the stability of the proposed family is conditional, so that the stability domain for each of the first three orders ( 5, 6, and 7) is smaller than that for the classic fourth-order Runge-Kutta method. However, as a positive point, the accuracy of the family boosts as the order of the family increases. As another positive point, any arbitrary order of the family can be easily achieved by solving the nonlinear algebraic equations. The robustness and ability of the authors’ schemes are illustrated over several useful time integration methods, such as Newmark linear acceleration, generalized-𝛼, and explicit and implicit Runge-Kutta methods. Moreover, various numerical experiments are utilized to show higher performances of the explicit family over the other methods in accuracy and computation time. The results demonstrate the capability of the new family in analyzing nonlinear systems with many degrees of freedom. Further to this, the proposed family achieves accurate results in analyzing tall building structures, even if the structures are under realistic loads, such as ground motion loads.

In this article, a method for solving the fractional Riccati differential equation is presented, which is based on the combination of Laplace transform and Runge-Kutta methods. In this way, first, by using the Laplace transform, the fractional derivative of Caputo in the fractional Riccati equation is converted into the ordinary derivative, and then, the resulting ordinary differential equation of the correct order is solved using the fourth-order Runge-Kutta method. Also, the error estimate and convergence are investigated. In addition, examples are provided to demonstrate the effectiveness of this method in practice. These examples show that the proposed method can give the approximate solution of fractional Riccati differential equations with high accuracy. Also, another advantage of the proposed method is that the approximate solution of fractional Riccati differential equations can be provided with appropriate accuracy in time intervals greater than one (maximum absolute errors 10^−5 over t ∈ [0, 8]). Additionally, the proposed Laplace-based reformulation removes the need to carry the full time-history of the solution, leading to a simpler time-domain model that is easier to handle in practice.