Solutions of nonlinear real world problems by a new analytical technique

Abstract

Here a new analytical scheme is presented to solve nonlinear boundary value problems (BVPs) of higher order occurring in nonlinear phenomena. This method is called second alternative of Optimal Homotopy Asymptotic Method. It converts a complex nonlinear problem into zeroth order and first order problem. A homotopy and auxiliary functions which are consisted of unknown convergence controlling parameters are used in this technique. The unknown parameters are determined by minimizing the residual. Many methods are used to determine these parameters. Here Galerkin's method is used for this purpose. It is applied to solve non-linear BVPs of order four, five, six, and seven. The Consequences are compared with other methods e.g., Differential Transform Method (DTM), Adomain Decomposition Method (ADM), Variational Iteration Method (VIM), and Optimal Homotopy Asymptotic Method (OHAM). It gives efficient and accurate first-order approximate solution. The achieved results are compared with the exact solutions as well as with other methods to authenticate the applied technique. This method is very simple and easy but more operative.

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

“…¼ 0: Combining Set 2 with Equations ( 7) and ( 15) together with elliptic functions from the previous table, one reach exact results of Equation (10) in the following:…”

“…where w represents wave velocity and δ 1 ; δ 2 ; δ 3 are wave numbers in the x; y; z directions, respectively. Accomplish Equation (10) with Equation ( 12) yields:…”

“…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].…”