Types Of Nonlinear Differential Equations Research Articles

Approximate analytical solutions to nonlinear oscillations via semi-analytical method

A modified algebraic method is proposed in this study as an effective semi-analytical method for obtaining analytical solutions for two types of nonlinear differential equations. The trial solution is assumed to be a polynomial, and the algebraic approach is used to find the unknown coefficients. To enhance the solution, first, we apply the Laplace transformation on the series solution and then create the Padé approximations of the resultant equation. Finally, for the nonlinear problems under discussion, the inverse Laplace transformation is used to obtain periodic solutions.

Modified VIM for wave equations

During the past few years, there has been a significant amount of interests in the Variational Iteration Method (VIM), which is a method that is utilized for the purpose of solving nonlinear differential equations. Minor adjustments have been made to ‘VIM as part of the Modified Variational Iterative Method (MVIM)’. For a specific type of nonlinear differential equation, the implementation of the MVIM method results in unnecessary calculations and additional time spent on repeated calculations for series solutions. The drawbacks of VIM are addressed by introducing a modified version, and its usefulness is demonstrated through a few examples.

On the Forms of Meromorphic Solutions of Some Type of Non-linear Differential Equations

On the Forms of Meromorphic Solutions of Some Type of Non-linear Differential Equations

Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects

Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects

Analytical technique for solving strongly nonlinear oscillator differential equations

The main motivation of this paper is to apply general formulas of the gamma function to find the analytical solutions of some types of nonlinear differential equations with recognizable simplicity and less calculation compared to other existing analytical methods in the literature. The simplicity and accuracy of the Gamma function method (GFM) in solving strongly nonlinear differential equations can be displayed through three practical examples arising in many different fields of engineering and science. The results obtained analytically are compared with the well-known analytical methods already published in the literature, such as Akbari-Ganji's method (AGM), homotopy perturbation method (HPM), and exact numerical solutions to confirm the efficiency of GFM. Excellent agreement is found between the analytical solutions and the exact numerical ones. Accordingly, the GFM is more reliable for exploring approximate solutions for strongly nonlinear oscillators. Moreover, the GFM can be easily extended to other nonlinear problems and is, therefore, very applicable in engineering and applied sciences.

Stability, Boundedness, and Square Integrability of Solutions of Neutral Fourth-Order Differential Equations

The purpose of this paper is to establish a new result, which guarantees the asymptotic stability and boundedness of the zero solution and the square integrability of solutions and their derivatives to neutral type nonlinear differential equations of fourth-order. We illustrate our results by an example at the end of the paper.

Indifference Computer Experiment for Mathematical Identification of Two Variables

In order to understand the two types of nonlinear differential equation problems in engineering dynamics, the author proposes a numerical analysis method for the two types of nonlinear differential equations based on computer simulation. This method establishes the MATLAB algorithm structure of the numerical solution of the fourth-order fixed-step Runge-Kutta and Lorenz models, discusses the error control in the case of variable step size, and plots the numerical solutions of the Lorenz system based on MATLAB in two-dimensional and three-dimensional space graphics. The x -direction displacement and y -direction displacement data are extracted from the Lorenz equation as iterative samples of the model, the regression curve obtained after iteration has a slope of 0.996, and the iterative regression model reflects the basic characteristics of the data well. This method presents the basic idea of numerical solution verification within acceptable error limits. For solving engineering problems with differential equations as mathematical models, an effective numerical solution method is provided, and further discussion on the numerical solutions of partial differential equations is of great significance.

A Galerkin–Newton Algorithm for Solution of a Kirchhoff-Type Static Equation

In this paper, an algorithm is proposed to find an approximate solution for the Kirchhoff -type nonlinear differential equation, which describes the static state of a beam. The solution of the problem consists of two parts. First, we apply the Galerkin method. Next, to solve the obtained discrete system of equations, we use the Newton iteration method. The algorithm total error is estimated. The results of the numerical experiment are given.

On distinctive solitons type solutions for some important nonlinear Schrödinger equations

The extended Jacobi elliptic function expansion (EJEFE) method is used to retrieve several types of optical solitons of two nonlinear Schrodinger equations, namely the Heisenberg ferromagnetic spin chains and Alfven envelop equations. The obtained traveling wave solutions and the corresponding plots are analysed by means of the symbolic package Mathematica. The solutions show that the proposed strategy is effective and reliable for solving different types of nonlinear differential equations.

Green’s Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems

In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

Optical solitons of space-time fractional Fokas\u2013Lenells equation with two versatile integration architectures

Nonlinear Schrödinger’s equation and its variation structures assume a significant job in soliton dynamics. The soliton solutions of space-time fractional Fokas–Lenells equation with a relatively new definition of local M-derivative have been recovered by utilizing improved tan (frac{phi (eta )}{2})-expansion method and generalized projective Riccati equation method. The obtained solutions are periodic, dark, bright, singular, rational, along with few forms of combo-soliton solutions. These solutions are given under constraints conditions which ensure their existence. The impact of local fractional parameter is featured by its graphical portrayal. 2D and 3D diagrams are drawn to illustrate the efficacy of the conformable fractional order on the behavior of some of those solutions. The secured solutions of this model have dynamic and significant justifications for some real-world physical occurrences. Our study shows that the suggested schemes are effective, reliable, and simple for solving different types of nonlinear differential equations.

A variety of new optical soliton solutions related to the nonlinear Schrödinger equation with time-dependent coefficients

A variety of soliton solutions of a common shape of the nonlinear Schrödinger equation with time-dependent coefficients is investigated via the extended tanh expansion method. This model was appealing to many researchers when its coefficients vary with time because they are closer to the naturalistic circumstances and they have a variety of soliton structures than the other classical ones (they able to model the slowly-changing inhomogeneities, external forces and non-uniformities of boundaries). A bundle of strands can be utilized to create a gadget called an endoscope or a fiberscope. Typically a long, lean test that can be put into a little gap, that will send a picture of what is interior through the fiber of a camera. Endoscopes are used by doctors to see inside the human body and are sometimes used by engineers to see inside tight spaces in machines. The four kinds of bizarre non-Kerr laws deliberated in this work are a quadratic-cubic law, an anticipates law, a cubic-quintic-septic law, and the triple-power law. Finally, 3D graphs are drawn for some acquired solutions. These solutions show that the suggested scheme is effective, reliable and simple for solving different types of nonlinear differential equations.

Meromorphic solutions of two certain types of nonlinear differential equations

We investigate the meromorphic solutions of two types of nonlinear differential equations of the form b f n + a f n − 1 f ′ + Q d ( f ) = u ( z ) e v ( z ) , n ≥ d + 2 , f n + f n − 1 f ′ + Q d ( f ) = P 1 ( z ) e α 1 + P 2 ( z ) e α 2 , n ≥ d + 3 , where a , b are constants with ( a , b ) ≠ ( 0 , 0 ) and n , d are positive integers, u , P 1 , P 2 are nonzero rational functions, v , α 1 , α 2 are nonconstant polynomials, and Q d ( f ) denotes a differential polynomial in f with rational functions as its coefficients. Our results improve some recent related results.

Semi-analytical study of Pine Wilt Disease model with convex rate under Caputo–Febrizio fractional order derivative

Abstract In this paper, we present semi-analytical solution to Pine Wilt disease (PWD) model under the Caputo–Fabrizio fractional derivative (CFFD). For the proposed solution, we utilize Laplace transform coupled with Adomian decomposition method abbreviated as (LADM). The concerned method is a powerful tool to obtain semi-analytical solution for such type of nonlinear differential equations of fractional order (FODEs) involving non-singular kernel. Furthermore, we give some results for the existence of solution to the proposed model and present numerical results to verify the established analysis.

Meromorphic Solutions of Certain Types of Non-linear Differential Equations

Let $$p_1, p_2$$ and $$\alpha _1, \alpha _2$$ be non-zero constants, and $$P_d(z, f)$$ be a differential polynomial in f of degree d. Li obtained the forms of meromorphic solutions with few poles of the non-linear differential equations $$f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}$$ provided $$\alpha _1\ne \alpha _2$$ and $$d\le n-2$$. In this paper, given $$d=n-1$$, we find the forms of meromorphic solutions with few poles of the above equations under some restrictions on $$\alpha _1, \alpha _2$$. Some examples are given to illustrate our results.

Entire solutions of certain type of non-linear differential equations

AbstractUtilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| > |α2| > |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

On entire solutions of certain type of nonlinear differential equations

In this paper, we shall extend some results regarding the growth estimate of entire solutions of certain type of linear differential equations to that of nonlinear differential equations. Moreover, our results will include several known results for linear differential equations obtained earlier as special cases.

On the Non-existence of Meromorphic Solutions of Certain Types of Non-linear Differential Equations

In this paper, we study meromorphic solutions of non-linear differential equations of the form $$f^n+P_d(f)=p_1e^{\alpha _1(z)}+p_2e^{\alpha _2(z)}$$ , where $$\alpha _1,\alpha _2$$ are polynomials of degree $$k(\ge 1)$$ , $$p_1$$ , $$p_2$$ are small meromorphic functions of $$e^{z^k}$$ , $$P_\mathrm{d}(f)$$ is a differential polynomial in f of degree d with small meromorphic functions of f as its coefficients. Some sufficient conditions on the non-existence of meromorphic solutions of such equations are provided. Our results complement some previous results.

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r, f) = S(r, f) of the following type of nonlinear differential equations: $${f^n} + {P_{n - 2}}\left( f \right) = {p_1}\left( z \right){e^{{\alpha _1}\left( z \right)}} + {p_2}\left( z \right){e^{{\alpha _2}\left( z \right)}},$$ where n ≥ 2 is an integer, Pn−2(f) is a differential polynomial in f of degree not greater than n−2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z) are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.

Analytical solutions of the Gerdjikov–Ivanov equation by using exp(−φ(ξ))-expansion method

The Gerdjikov–Ivanov equation which appears in quanta field theory, nonlinear optics, weak nonlinear dispersive water wave, etc., is discussed. We study a new direct approach namely exp(−φ(ξ))-expansion method to seek exact solutions of Gerdjikov–Ivanov equation which include variety of models. These exact solutions include the hyperbolic, trigonometric, rational and exponential functions with some arbitrary parameters. It is shown that the method is quite efficient and useful to obtain exact solutions of different types of nonlinear differential equations.