Separation of variables in the nonlinear wave equation

<abstract><p>In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.</p></abstract>

Solution of nonlinear partial differential equations from base equations

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

A Conservative Spectral Method for Several Two-Dimensional Nonlinear Wave Equations

A first, we used our knowledge of Fourier series to solve several interesting boundary value problems by the method of separation of variables. The success of our method depended to a large extent on the fact that the domains under consideration were easily described in Cartesian coordinates. In this paper/research we address problems where the domains are easly described in polar and cylindrical coordinates. Spesifically we consider boundary value problems for the wave, heat, Laplace and Poisson equation over disks or cylinders. Upon restating these problems in suitable coordinat systems and separating variables, we will encounter new ordinary differential equations, Bessel’s equation, whose solutions are called Bessel function in ways analogous to Fourier series expansions. The vibrations of the membrane are governed by the two-dimensional wave equation, which will be expressed in polar coordinantes, because these are the coordinates best suited to this problem. Finally, we will solve the two dimensional wave equation in polar coordinates (general case).

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

We develop a generalized conditional symmetry approach for the functional separation of variables in a nonlinear wave equation with a nonlinear wave speed. We use it to obtain a number of new (1+1)-dimensional nonlinear wave equations with variable wave speeds admitting a functionally separable solution. As a consequence, we obtain exact solutions of the resulting equations.

Variable separation method for a nonlinear time fractional partial differential equation with forcing term

This paper is concerned with the lifespan of and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation ∂ t 2 u − ∑ i = 1 2 ∂ i ( c i 2 ( u ) ∂ i u ) \partial _t^2u-\sum _{i=1}^2\partial _i(c_i^2(u)\partial _iu) = 0 =0 , where c i ( u ) ∈ C ∞ ( R n ) c_i(u)\in C^{\infty }(\mathbb {R}^n) , c i ( 0 ) ≠ 0 c_i(0)\neq 0 , and ( c 1 ′ ( 0 ) ) 2 + ( c 2 ′ ( 0 ) ) 2 ≠ 0 (c_1’(0))^2+(c_2’(0))^2\neq 0 . This equation has an interesting physical background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition ( u ( 0 , x ) , ∂ t u ( 0 , x ) ) = ( ε u 0 ( x ) , ε u 1 ( x ) ) (u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x)) with u 0 ( x ) , u 1 ( x ) ∈ C 0 ∞ ( R 2 ) u_0(x), u_1(x)\in C_0^{\infty }(\mathbb {R}^2) , and ε > 0 \varepsilon >0 is small, we will show that the classical solution u ( t , x ) u(t,x) stops to be smooth at some finite time T ε T_{\varepsilon } . Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives ∇ t , x u ( t , x ) \nabla _{t,x}u(t,x) , while u ( t , x ) u(t,x) itself is continuous up to the blowup time T ε T_{\varepsilon } .

The purpose of this paper is to extend the store of models able to support integrable defects by investigating the two-dimensional Boussinesq nonlinear wave equation. As has been previously noted in many examples, insisting that a defect contributes to energy and momentum to ensure their conservation, despite the presence of discontinuities and the explicit breaking of translation invariance, leads to sewing conditions relating the two fields and their derivatives on either side of the defect. The manner in which several types of soliton solutions to the Boussinesq equation are affected by the defect is explored and reveals new effects that have not been observed in other integrable systems, such as the possibility of a soliton reflecting from a defect or of a defect decaying into one or two solitons.

The Lie group method is a powerful technique for obtaining analytical solutions for various nonlinear differential equations. This study aimed to explore the behavior of nonlinear elastic wave equations and their underlying physical properties using Lie group invariants. We derived eight-dimensional symmetry algebra for the (3+1)-dimensional nonlinear elastic wave equation, which was used to obtain the optimal system. Group-invariant solutions were obtained using this optimal system. The same analysis was conducted for the damped version of this equation. For the conservation laws, we applied Noether’s theorem to the nonlinear elastic wave equations owing to the availability of a classical Lagrangian. However, for the damped version, we cannot obtain a classical Lagrangian, which makes Noether’s theorem inapplicable. Instead, we used an extended approach based on the concept of a partial Lagrangian to uncover conservation laws. Both techniques account for the conservation laws of linear momentum and energy within the model. These novel approaches add an application of variational calculus to the existing literature. This offers valuable insights and potential avenues for further exploration of the elastic wave equations.

In the current paper, a simplified model of Tower Cranes has been presented in order to investigate and analyze the nonlinear differential equation governing on the presented system in three different cases by Algebraic Method (AGM). Comparisons have been made between AGM and Numerical Solution, and these results have been indicated that this approach is very efficient and easy so it can be applied for other nonlinear equations. It is citable that there are some valuable advantages in this way of solving differential equations and also the answer of various sets of complicated differential equations can be achieved in this manner which in the other methods, so far, they have not had acceptable solutions. The simplification of the solution procedure in Algebraic Method and its application for solving a wide variety of differential equations not only in Vibrations but also in different fields of study such as fluid mechanics, chemical engineering, etc. make AGM be a powerful and useful role model for researchers in order to solve complicated nonlinear differential equations.

Modeling and simulation of water infiltration from the ground surface into unsaturated–saturated soil is very important in the study of the hydrological cycle process. Transient fluid flow through unsaturated–saturated soil is generally described by Richards’ equation. The equation is a nonlinear parabolic partial differential equation for which many numerical and limited analytical solutions exist. In this paper, exponential models for water content and hydraulic conductivity are selected so the nonlinear equation becomes linear and then separation of variables and Fourier series expansion techniques are used to derive analytical solutions to 2D water infiltration into unsaturated–semi-saturated soils with different distributions of water content on a part and/or parts of the top boundary. 2D water infiltrations are investigated in a rectangular soil domain, where high water content zone is considered as semi-saturated zone and residual water content zone is considered as unsaturated zone at initial time. A total of three cases are investigated in this paper. In case 1, a uniform distribution of water content on a part at the top boundary and no-flow boundaries at side edges are considered. Also, horizontal water table in unsaturated–semi-saturated domain is assumed as initial condition. In case 2, an inclined water table initial condition and a uniform distribution of water content on a part at the top boundary are investigated. Finally, in case 3, an analytical solution for ramp distributions of water content on two parts at the top boundary and an inclined water table initial condition is obtained. To illustrate the use of the derived equations, water content values are obtained from a numerical solution and compared to those from analytical solutions for two cases, showing less than 2% errors. These analytical solutions may be used as a benchmark for verification and efficiency assessment of numerical approaches.

By using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution. Copyright © 2015 John Wiley & Sons, Ltd.

A possibility for predicting the time-dependent pattern of sickness of human beings, i.e., biological catastrophe, has been shown by proposal of a nonlinear ordinary differential equation, describing temporal features of six macroscopic molecular groups chemically interacting in living beings, (Naitoh in Jpn J Ind Appl Math 28:15–26, 2011; Naitoh and Inoue in J Artif Life Robot 18:127–132, 2013). Logically, we also find that, six is minimum and essential as the number of macroscopic molecular groups for describing living systems. Then, along with the number theory applied for the differential equation, we derive critical mathematical conditions for predicting the premonition just before sickness (discrepancy from a healthy condition), which agree with an important knowledge revealed by the linear analysis proposed by Chen (Dynamical network biomarkers for identifying early-warning signals of complex diseases, Beppu, Oita Japan, 2015). In the present report, we first show that computational several time-histories of sickness obtained by solving the nonlinear differential equation with various parameters describing polymorphism agree well with actual time-dependent patterns of sickness for human beings, which is further evidence of usefulness of the nonlinear differential equation and its critical mathematical conditions. Thus, to examine the boundary between biological and abiological chemical reaction systems, which is related to the origin of living systems, we next check whether or not the nonlinear equation can also predict such abiological catastrophic phenomenon as misfire in artifacts including mechanical combustion engines.