An Iterative Method for Solving Inverse Problems of a Nonlinear Wave Equation

An iterative numerical algorithm for solving multi-parameter inverse problems of evolutional partial differential equations

This paper presents results of an investigation of the driving of finite amplitude oscillations in closed constant diameter ducts and ducts whose cross sectional area varies with axial distance. Finite amplitude oscillations are of interest in acoustic compression and thermoacoustics. The behavior of these oscillations was modeled by a non-linear wave equation in terms of velocity potential that accounted for second order nonlinearities, dissipation and periodic energy addition. Solutions were obtained using the Galerkin method, which yields considerable insight into the nature of the oscillations. The results of this study show that significantly higher amplitude pressure oscillations (-2.5 times) can be forced in horn-like shaped ducts as opposed to cylindrical ducts. Furthermore, highly dissipative shock-like waveforms are excited in the constant diameter ducts in contrast with practically continuous waveforms excited in horn-like ducts. Shock-like waveforms excited in constant diameter ducts are caused by the generation of higher harmonics through efficient non-linear coupling with the fundamental mode. In contrast, the non-linear coupling between the fundamental mode and its' harmonics is weak in ducts whose cross sectional area varies axially (e.g., the investigated horn-like ducts), which minimizes the excitation of harmonics in such ducts. This concentrates the forced power input in the fundamental mode, resulting in the forcing of continuous waveform, large amplitude oscillations. This paper has also demonstrated that the application of the Galerkin method to the solution of this non-linear problem can yield physical insight into the physics of the problem at a low computational cost.

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

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 } .

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.

We consider in this paper the 1-dim nonlinear wave equation $$\frac{\partial ^2 u}{\partial t^2}(t,x)=\frac{\partial ^2 u}{\partial x^2}(t,x)+|u|^{1+\alpha }(t,x)\;(\alpha >0)$$ and its finite difference analogue. It is known that the solution of the current equation may become unbounded in finite time, a phenomenon which is known as blow-up. Moreover, since the nonlinear wave equation enjoys finite speed of propagation, even if the solution has become unbounded at certain points, the solution continues to exist, blows up at later times and forms the so-called blow-up curve. Up to the present, however, numerical approximation for blow-up problems gave only the convergence up to the blow-up time of $$\Vert u(t,\cdot )\Vert _\infty $$ . In fact, adaptive time mesh strategy was used to numerically reproduce the phenomenon of finite-time blow-up. Nevertheless, computation using such schemes can not be extended beyond the blow-up time of $$\Vert u(t,\cdot )\Vert _\infty $$ . This is a fatal problem when computing blow-up solutions for the nonlinear wave equation. As a consequence, we reconsider a finite difference scheme whose temporal grid size is given uniformly and propose an algorithm with rigorous convergence analysis to numerically reconstruct the blow-up curve for the nonlinear wave equation.

This paper is concerned with a class of nonlinear stochastic wave\nequations in $\\mathbb{R}^d$ with $d \\leq 3$, for which the nonlinear terms are\npolynomial of degree $m$. As an example of the nonexistence of a global\nsolution in general, it is shown that there exists an explosive solution of\nsome cubically nonlinear wave equation with a noise term. Then the existence\nand uniqueness theorems for local and global solutions in Sobolev space $H_1$\nare proven with the degree of polynomial $m \\leq 3$ for $d = 3$, and $m \\geq 2$\nfor $d = 1$ or 2.

This chapter is devoted to an essential extension of the finite-energy condition for extended Runge–Kutta–Nystrom (ERKN) integrators when applied to nonlinear wave equations. We begin with an error analysis of ERKN integrators for multi-frequency highly oscillatory systems \(y''+My=f(y)\), where M is positive semi-definite, \(\Vert M\Vert \gg max\{1,\Vert \frac{\partial f}{\partial y}\Vert \}\). These highly oscillatory problems arise from the semi-discretisation of conservative or dissipative nonlinear wave equations. The structure of M and the initial conditions are dependent on the particular spatial discretisation. A finite-energy condition for the semi-discretisation of nonlinear wave equations is introduced and analysed. This is similar to the error analysis for Gaustchi-type methods of order two, where a finite-energy condition bounding amplitudes of high oscillations is satisfied by the solution. These ensure that the error bound for ERKN methods is independent of \(\Vert M\Vert \). Since stepsizes are not restricted by frequencies of M, large stepsizes can be employed by our ERKN integrators which may be of arbitrarily high order. The numerical experiments presented in this chapter demonstrate that our results are really promising, and consistent with our analysis and predictions.

Internal resonance induced nonlinear vibration and acoustic radiation of a hyperelastic cantilever structure immersed in fluid

Nonlinear vibration and acoustic radiation of an internally resonant buckled beam

The well-known theory for energy dissipation in weak shock waves is applied to the case of propagating and standing acoustic waves of finite amplitude. The known results for the decay of plane, cylindrical, and spherical waves propagating in a fluid are obtained by means of a simple technique previously used by Rudnick [J. Acoust. Soc. Amer. 25, 1012 (L) (1953)] and based on the assumption that the waves are of the sawtooth type with a weak discontinuity each wavelength. The same technique is then applied to large-amplitude resonant oscillations in tubes, assuming that the acoustic field can be represented by two sawtooths traveling in opposite directions inside the tube. These calculations are of a very simple nature and yield solutions that compare favorably with results obtained by more rigorous techniques.

Multimodal data, quantified multidimensionally through characteristics obtained from images of plant posture and environmental data, included environmental conditions and growth states affecting the growth and development periods of plants, is expected to gradually change. The Sliding Window-based Support Vector Regression (SW-SVR) allowed development of a model that used a wide variety of complex plant physiological information, realistic measurement data and machine learning with time granularity, to create a soft sensor for wilt detection in plants. The SW-SVR allowed development of a model that used a wide variety of complex plant physiological information, realistic measurement data and machine learning with time granularity, to create a soft sensor for wilt detection in plants. Visual characteristics relating to plant wilting are extracted, using a Convolutional Neural Network, which is one type of deep learning that has been effective in its application to the field of image recognition.

A new widly convergent method for solving the problem of operator identification is illustrated. Numerical simulations are carried out to test the feasibility and to study the general characteristics of the technique without the real measurement data. This technique is a direct application of the continuation homotopy method for solving nonlinear systems of equations. It is found that this method does give excellent results in solving the inverse problem of the elliptic differential equations.

Global dynamics of a pipe conveying pulsating fluid in primary parametrical resonance: Analytical and numerical results from the nonlinear wave equation

A corrected derivation of nonlinear wave propagation equations with fractional loss operators is presented. The fundamental approach is based on fractional formulations of the stress-strain and heat flux definitions but uses the energy equation and thermodynamic identities to link density and pressure instead of an erroneous fractional form of the entropy equation as done in Prieur and Holm ["Nonlinear acoustic wave equations with fractional loss operators," J. Acoust. Soc. Am. 130(3), 1125-1132 (2011)]. The loss operator of the obtained nonlinear wave equations differs from the previous derivations as well as the dispersion equation, but when approximating for low frequencies the expressions for the frequency dependent attenuation and velocity dispersion remain unchanged.