Modified and new iterative method in solving linear and nonlinear differential equations

Abstract

Differential equations are encountered in various fields such as physics, chemistry, biology, mathematics, and engineering. Therefore, there has been much attention to searching for better and more efficient techniques for finding approximate or exact, analytical or numerical solutions, to linear and nonlinear models. In general, the semi analytical or iterative methods effectively find numerical and explicit series solutions. One of these methods is the New Iterative Method (NIM) or Daftardar-Jafari iterative method (DJM). This study presents a new general framework of NIM to solve initial value problems (IVPs), boundary value problems (BVPs), as well as system of ordinary differential equations (ODEs). A systematic literature review of NIM has been done to ensure that selection of mathematical models in this study has not yet been solved using NIM. From the observations based on the literature review, it is hopeless to find analytical or numerical solutions for a class of infectious disease models' valid globally for long-time span. To overcome this shortcoming, a new technique based on the NIM, combining the standard NIM and Laplace Transform called as Modified New Iterative Method (MNIM) has been proposed for the first time to solve a class of nonlinear system of OD Es. The numerical precisions of NIM and MNIM are compared with the well-known purely numerical fourth-order Runge-Kutta method (RK4) and some other benchmarking schemes available in the literature. The continuous step time M = 0.001 has been used for RK4 for all computation works. In particular, the solutions of NIM for linear IVPs of lake pollution model and nonlinear population dynamics model indicate accurate result that quickly converges to the RK4 solutions with 9-iterations and 5-iterations involved, respectively. Besides that, 6-iterations of NIM solutions has been used for the chemical kinetics system, diffusionless Lorenz system and Rossler system. Furthermore, the NIM also tested to BVPs of heat equations and found to give noticeable result using only 3-iterations. The MNIM solved Human Immunodeficiency Virus (HIV) infection model and Susceptible-Infected-Recovered (SIR) epidemic model using 2-iterations and 3-iterations, respectively. The results show that the solutions by MNIM match with those of RK4 at least 8-decimal places for HIV model. In case of SIR model, highly precise approximation has been achieved as both the RK4 and MNIM precisions coincide each other in time interval [O, I 00]. In contrast, the 5- iterations NIM solutions valid for only a small time span as compared to the solution of RK4 and MNIM has been used to extend the validity domains for selected disease models. The correctness of the MNIM is found much better than NIM and the other existing solutions. The standout feature of MNIM is that it requires less computation steps, yet it can provide high accuracy without any linearization or discretization. The advantage of the NIM or MNIM is that it does not require a multiplier or any polynomials for nonlinear terms of the problems, which makes it easier in solving the nonlinear problems. As a result, very simple solution procedures with decent accuracy are found that emphasize the reliability and wider applicability of both the NIM and MNIM techniques in the linear and nonlinear models. Thus, it can be concluded that the proposed iterative techniques are convenient and efficient for solving linear and nonlinear problems arising in science and engineering.