A new multistage technique for approximate analytical solution of nonlinear differential equations

Abstract

The article introduces a new multistage technique for solving a polynomial system of nonlinear initial and boundary value problems of differential equations. The radius of convergence R of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Then guided by the convergence-control parameter , the domain of the problem is split into subintervals. By stepping out in a multistage manner, corresponding subproblems are defined which are then subsequently solved with conventional Parker-Sochacki method to get a piecewise continuous solution with very high accuracy. The method is applied to SIR epidemic model, stiff differential equation modelling combustion, Lorenz chaotic problem, and the Troesch's boundary value problem. The results obtained showed a remarkable accuracy when compared with Runge-Kutta Method of order 4. The article showcased the proposed method as a simple, yet accurate approximate analytical technique for nonlinear differential equations.

Direct numerical solutions of the SIR and SEIR models via the Dirichlet series approach

“…In this step we get u N ðtÞ ¼ P N i¼0 a i e À il 1 t . Step 5: Plotting S N (t), I N (t), R N (t) by using Eqs (24)(25)(26) We first show manually how to get some c n by performing Step 1…”

“…The power series solution of the SIR model has been studied by [24]. A multistage technique repeating the same order of power series approximation with updated initial conditions is a method to get a numerical solution of the SIR model [25]. Using power series approximation or repeatedly using a fixed order of power series approximation can not eventually reach the long-run behavior of the system due to the divergence of the power series.…”

Ion Acoustic Solitary Wave Solutions to mKdV-ZK Model in Homogeneous Magnetized Plasma

“…Therefore, the study of accurate exact solutions to realize internal properties of NEMs demonstrates a crucial task for understanding of the majority nonlinear substantial happenings or acquiring novel occurrences. More scientists expended efforts to release inventive proficient procedures designed for explanation of interior properties of NEMs amid with constant coefficients has been established such as a distinct arithmetic structure [5], IRM-CG technique [6], transformed rational function method [7], fractional residual technique [8], new multistage procedure [9], new analytical method [10], extended tanh scheme [11], Hirota-bilinear scheme [12][13][14], multi expexpansion technique [15,16], the Kudryashov and the extended sine-Gordon schemes [17], variable separation method [18], MSE method [19], the nonlinear capacity [20], Jacobi elliptic function [21], the pitchfork bifurcation [22], ansatz method [23], higher order rogue wave [24], fractional natural decomposition method [25], generalized exponential rational function scheme [26], numerical and three analytic schemes [27], 1=G 0 -expansion scheme [28], the φ 6 -model method [29,30], and so on. All of the above techniques provide sinusoidal and hyperbolic results.…”

Abundant Bounded and Unbounded Solitary, Periodic, Rogue‐Type Wave Solutions and Analysis of Parametric Effect on the Solutions to Nonlinear Klein–Gordon Model

“…Many scientific experimental models are employed in nonlinear differential form from the phenomena of nonlinear fiber optics, high-amplitude waves, fluids, plasma, solid state particle motions, etc. Surveying literature, we realized ideas that many scientists worked to disclose innovative, efficient techniques for explaining internal behaviors of NLDEs with constant coefficients that are significant to elucidate different intricate problems such as a discrete algebraic framework [1], IRM-CG method [2], transformed rational function scheme [3], fractional residual method [4], new multistage technique [5], new analytical technique [6], extended tanh approach [7], Hirota-bilinear approach [8][9][10], multi exp-expansion method [11,12], Jacobi elliptic expansion method [13,14], Lie approach [15], Lie symmetry analysis techniques [16], generalized Kudryashov scheme [17,18], generalized exponential rational function scheme [19], MSE method [20][21][22], and many more. Such or similar schemes are also used to solve the model with variable coefficients to visualize various new nonlinear dynamics [23][24][25].…”