In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the step size ratio r is left the same, halved, or doubled in order to optimize the total number of steps, minimize the number of formulae stored in the code, and ensure that the method is zero-stable. The method is formulated by integrating the Lagrange polynomial with limits of integration selected at special points. The article further analyzed the stability, order, consistency, and convergence properties of the VSHBM. The stability regions of the VSHBM at different values of the step size ratios were also plotted and plots showed that the method is fit for solving the Kepler problem. The results generated were then compared with some existing methods, including the MATLAB inbuilt stiff solver (ode 15 s), with respect to total number of failure steps, total number of steps, total function calls, maximum error, and computation time.

This paper aims to present two nonstandard finite difference (NFSD) methods to solve an SIR epidemic model. The proposed methods have important properties such as positivity and boundedness and they also preserve conservation law. Numerical comparisons confirm that the accuracy of our method is better than that of other existing standard methods such as the second-order Runge–Kutta (RK2) method, the Euler method and some ready-made MATLAB codes.

We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space ${\mathbb{R}}^m$. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability of the solutions.

Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the Lagrange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include order, consistence, zero-stability, convergence, and region of absolute stability were further carried out. The algorithm was then applied to solve some samples mildly and highly stiff differential systems and the results generated were compared with those of some existing methods in terms of the total number of steps taken, number of function evaluation, number of failure/rejected steps, maximum errors, absolute errors, approximate solutions and execution time. The results obtained clearly showed that the NFSSA performed better than the existing ones with which we compared our results including the inbuilt MATLAB stiff solver, ode 15s. The results were also computationally reliable over long intervals and accurate on the abscissae points which they step on.

Second-order oscillatory problems have been found to be applicable in studying various phenomena in science and engineering; this is because these problems have the capabilities of replicating different aspects of the real world. In this research, a new hybrid method shall be formulated for the simulations of second-order oscillatory problems with applications to physical systems. The proposed method shall be formulated using the procedure of interpolation and collocation by adopting power series as basis function. In formulating the method, off-step points were introduced within the interval of integration in order to bypass the Dahlquist barrier, improve the accuracy of the method and also upgrade the order of consistence of the method. The paper further validated the some properties of the hybrid method derived and from the results obtained; the new method was found to be consistent, convergent and stable. The simulation results generated as a result of the application of the new method on some second-order oscillatory differential equations also showed that the new hybrid method is computationally reliable.