A Final Value Problem for the Nonlinear Modified Helmholtz Equation Associated with the Nonlinear Wave Velocity

In the present paper explores a reverse Cauchy problem for a heat transfer issue described by the Helmholtz and modified Helmholtz equation. Our goal is to identify an unknown defect within a simply connected bounded domain , given the Dirichlet data (temperature) on the boundary , and Neumann data (heat flux) on the boundary . We assume that the temperature satisfies the Helmholtz equation (or modified Helmholtz equation) that governs the heat condition in the fin. To solve this problem, we propose a method that involves two steps. First, we solve a Cauchy problem using the Helmholtz equation (or modified Helmholtz equation) to determine the temperature Then, in the second phase, we solve a system of nonlinear scalar equations to determine the coordinates of the points defining the boundary . This can be achieved using an iterative method, such as Newton's method.

In this paper, we present the application of our recently developed Green's function Monte Carlo algorithm to the solution of the one-dimensional Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional Green's function Monte Carlo approach for the solution of partial differential equations subjected to Neumann and mixed boundary conditions involves “reflecting boundaries” resulting in relatively large computational times. Our algorithm, motivated by the work of K. K. Sabelfeld is philosophically different in that there is no requirement for reflection at these boundaries. The underlying feature of this algorithm is the elimination of the use of reflecting boundaries through the use of novel Green's functions that mimic the boundary conditions of the problem of interest. In the past, we have applied it to the solution of the one-dimensional Laplace equation and the modified Helmholtz equation. In this work, we apply it to the solution of the Helmholtz equation. In the case of the Helmholtz equation, unlike the Laplace equation and modified Helmholtz equation, the algorithm is constrained to quarter-wavelength length scales, a constraint that is the result of resonance in the Green's function for the Helmholtz equation. This constraint is also present in the case of the Helmholtz equation subjected to Dirichlet conditions and is not specific to Neumann and mixed boundary conditions. However, within this constraint, excellent agreement has been obtained between an analytical solution and numerical results.

In this work, it is shown that the geometry of a gravity field generated by a spheroid with low eccentricity can be described with the help of a newly modified Helmholtz equation. To distinguish this equation from the modified Helmholtz equation, we refer to it as the G-modified Helmholtz equation. The use of this new equation to study the spheroid’s gravity field is advantageous in expressing the gravity vector as a vector series of spherical harmonics. The solution of the G-modified Helmholtz equation involves both the gravity intensity g (or simply gravity g) and the intensity E of an electrostatic field as shown in sequel. An electrostatic field generated by an oblate spheroid charged with l electrons (uniform ellipsoidal charge distribution) is demonstrated to be a special case. Both gravity intensity g and intensity E are governed by the same law and can be expressed as a series of spherical harmonics, and thus the G-modified Helmholtz equation is useful for describing the gravity and electrostatic fields.

Performance assessment of internal porous structures on liquid sloshing in various 3D tanks by multi-domain IGABEM

Higher order three-dimensional fundamental solutions to the Helmholtz and the modified Helmholtz equations

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed methods.

In this paper we present a mesh-free ap- proach to numerically solving a class of second order time dependent partial differential equations which in- clude equations of parabolic, hyperbolic and parabolic- hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to stan- dard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equa- tions, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a se- quence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically com- puting a particular solution for the inhomogeneous mod- ified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approxi- mated by radial basis functions, polynomial or trigono- metric functions. Analytic particular solutions are pre- sented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solu- tion is subtracted off. Two types of Trefftz bases are con- sidered, F-Trefftz bases based on the fundamental solu- tion of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Var- ious techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented il- lustrating the accuracy and efficacy of this methodology.

Many Engineering Problems could be mathematically described by FinalValue Problem, which is the inverse problem of InitialValue Problem. Accordingly, the paper studies the final value problem in the field of ODE problems and analyses the differences and relations between initial and final value problems. The more general new concept of the endpoints-value problem which could describe both initial and final problems is proposed. Further, we extend the concept into inner-interval value problem and arbitrary value problem and point out that both endpoints-value problem and inner-interval value problem are special forms of arbitrary value problem. Particularly, the existence and uniqueness of the solutions of final value problem and inner-interval value problem of first order ordinary differential equation are proved for discrete problems. The numerical calculation formulas of the problems are derived, and for each algorithm, we propose the convergence and stability conditions of them. Furthermore, multivariate and high-order final value problems are further studied, and the condition of fixed delay is also discussed in this paper. At last, the effectiveness of the considered methods is validated by numerical experiment.

A new Green's function Monte Carlo algorithm for the estimation of the derivative of the solution of Helmholtz equation subject to Neumann and mixed boundary conditions

It is known that the nonlinear nonhomogeneous backward Cauchy problem ut(t) + Au(t) = f (t, u(t)), 0 ⩽ t < τ with u(τ) = φ, where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f, that a solution of the above problem satisfies an integral equation involving the spectral representation of A, which is also ill-posed. Spectral truncation is used to obtain regularized approximations for the solution of the integral equation, and error analysis is carried out with exact and noisy final value φ. Also stability estimates are derived under appropriate parameter choice strategies. This work extends and generalizes many of the results available in the literature, including the work by Tuan (2010) for linear homogeneous final value problem and the work by Jana and Nair (2016b) for linear nonhomogeneous final value problem.

The inverse Cauchy problems for elliptic equations, such as the Laplace equation, the Poisson equation, the Helmholtz equation and the modified Helmholtz equation, defined in annular domains are investigated. The outer boundary of the annulus is imposed by overspecified boundary data, and we seek unknown data on the inner boundary through the numerical solution by a spring-damping regularization method and its Lie-group shooting method (LGSM). Several numerical examples are examined to show that the LGSM can overcome the ill-posed behavior of inverse Cauchy problem against the disturbance from random noise, and the computational cost is very cheap.

Nonlinear science has been an important symbol in the development of modern science, especially the researches in nonlinear dynamics and nonlinear waves have extraordinary significance in solving the complex phenomena and problems encountered in various fields of natural science. In this paper, the nonlinear bending wave propagation of a piezoelectric laminated beam with electrical boundary conditions is studied. Firstly, considering the geometric nonlinear effect and piezoelectric coupling effect, the nonlinear equation of the one-dimensional infinite rectangular piezoelectric laminated beams is established by using Hamiltonian principle. Secondly, the Jacobi elliptic function expansion method is used to treat the nonlinear flexural wave equation, and the corresponding shock wave solution and solitary wave solution of the nonlinear flexural wave equation are obtained in the approximate case. Last, the nonlinear Schrodinger equation is obtained by using the reduced perturbation method, and the bright and dark soliton solutions are further obtained. Moreover, the effects of external voltage and the thickness of the piezoelectric layer on the characteristics of shock wave and solitary wave as well as bright and dark solitons are studied. The results show that when the wave velocity is small, the external voltage has a great influence on the shock wave, and when the wave velocity is large, the external voltage has no effect on the solitary wave. The bright solitons and the dark solitons can be obtained by adjusting the external voltage applied to the piezoelectric laminated beam. It is found that the amplitudes of bright and dark solitons increase with the increase of external voltages.

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over‐specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45–52) applied to the Cauchy problem for the two‐dimensional modified Helmholtz equation. The two mixed, well‐posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross‐validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two‐dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

Integral representations for the solutions of the Laplace and modified Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the solution and its normal derivative on the boundary, and for a well-posed boundary-value problem (BVP) one of these functions is unknown. Determining the Neumann data from the Dirichlet data is known as constructing the Dirichlet-to-Neumann map. A new transform method was introduced in Fokas (1997, Proc. R. Soc. Lond. A, 53, 1411–1443) for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs). For linear PDEs this method can be considered as the analogue of the Green's function approach in the Fourier plane. In this method the Dirichlet-to-Neumann map is characterized by a certain equation, the so-called global relation, which is formulated in the complex k-plane, where k denotes the complex extension of the spectral (Fourier) variable. Here we solve the global relation numerically for the Laplace and modified Helmholtz equations in a convex polygon. This is achieved by evaluating the global relation at a properly chosen set of points in the spectral (Fourier) plane, which is why this method has been called a ‘spectral collocation method’. Numerical experiments suggest that the method inherits the order of convergence of the basis used to expand the unknown functions, namely, exponential for a polynomial basis such as Chebyshev, and algebraic for a Fourier basis. However, the condition number of the associated linear system is much higher for a polynomial basis than for a Fourier one.

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert’s pseudo-charge method [Weinert M, J Math Phys, 1981, 22:2433–2439] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert’s convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.