Nonhomogeneous Wave Equation Research Articles

Solving Fuzzy Non-homogeneous Wave Equations via Laplace Residual Power Series Approach

Solving Fuzzy Non-homogeneous Wave Equations via Laplace Residual Power Series Approach

Boundedness of pseudo-differential operators via coupled fractional Fourier transform

In this article, we obtained some fruitful results of the coupled fractional Fourier transform and its kernel. We defined pseudo-differential operators related to coupled fractional Fourier transform on Schwartz spaces and it is shown that their composition is again a pseudo-differential operator. It made further discussion on the composition of two pseudo-differential operators on Sobolev spaces and derived certain norm inequalities on it. Further, we have successfully applied some of the results of the coupled fractional Fourier transform to investigate the solution of nth-order linear non-homogeneous partial differential equation and wave equation. Lastly, we present some examples involving graphs and tables to illustrate the validity of our theoretical findings.

Research on finite element model of rotor aerodynamicnoise in hovering state

The helicopter has been widely used in many fields, and the rotor aerodynamic noise has gradually attracted attention. The non-homogeneous wave equation, Ffowcs Williams-Hawkings equation (FW-H equation) derived from hydrodynamic N-S equation, combined with RANS turbulence simulation, was used to simulate the sound field of NACA0012 airfoil-shape model in hovering state. On this basis, by changing the thickness and camber of the blade, the aerodynamic noise of the rotor under different airfoil profile thickness and camber is calculated. The results show that the change of airfoil thickness and camber will affect the aerodynamic noise of the rotor in hovering state.

Innovative Method for Computing Approximate Solutions of Non-Homogeneous Wave Equations with Generalized Fractional Derivatives

In this work, a well-known non-homogeneous wave equation with temporal fractional derivative is approximately investigated. A recently defined generalized non-local fractional derivative is utilized as the fractional operator. A novel technique is proposed to approximate the solutions of wave equation with generalized fractional derivative. The proposed method is based on the shifted Chebyshev polynomials and a combination of collocation and residual function methods. Theoretical analysis of the convergence of the proposed method is performed. Approximate solutions are derived in both rectangular and non-rectangular (general) domains.

A New Method for Solving Sequential Fractional Wave Equations

In this article, we focus on two classes of fractional wave equations in the context of the sequential Caputo derivative. For the first class, we derive the closed-form solution in terms of generalized Mittag–Leffler functions. Subsequently, we consider a more general class of nonhomogeneous fractional wave equations. Due to the complexity of finding exact solutions for these problems, we employ a numerical technique based on the operational matrix method to approximate the solution. We provide several theoretical and numerical examples to validate the effectiveness of this numerical approach. The results demonstrate the accuracy and efficiency of the proposed method.

Energy-conserving successive multi-stage method for the linear wave equation with forcing terms

We propose a high-order time-discretized method for a non-homogeneous linear wave equation with a forcing term. The method conserves the accumulated discrete energy with the external term. We provide detailed proofs of unique solvability and unconditional energy conservation of the proposed successive multi-stage (SMS) method. We also present reduced order conditions up to the fourth order with aid of some important algebraic identities from the features of the SMS methods. We demonstrate the accuracy and stability of the SMS methods using numerical experiments. In addition, to show the applicability of the proposed method, we extend the method to solve quasi-linear wave equations and provide numerical simulations for sine-Gordon and Boussinesq-type equations.

Total Controllability for a Class of Fractional Hybrid Neutral Evolution Equations with Non-Instantaneous Impulses

This study demonstrates the total control of a class of hybrid neutral fractional evolution equations with non-instantaneous impulses and non-local conditions. The boundary value problem with non-local conditions is created using the Caputo fractional derivative of order 1<α≤2. In order to create novel, strongly continuous associated operators, the infinitesimal generator of the sine and cosine families is examined. Additionally, two approaches are used to discuss the solution’s total controllability. A compact strategy based on the non-linear Leray–Schauder alternative theorem is one of them. In contrast, a measure of a non-compactness technique is implemented using the Sadovskii fixed point theorem with the Kuratowski measure of non-compactness. These conclusions are applied using simulation findings for the non-homogeneous fractional wave equation.

Numerical solutions of the fuzzy wave equation based on the fuzzy difference method

Our study introduces, discusses, and investigates approximate solutions to the fuzzy Wave equation on a finite domain utilizing generalized Hukuhara partial differentiability. A computationally effective algorithm based on the fuzzy finite difference is proposed. In order to obtain fuzzy finite difference formulas, we extend a full fuzzy Taylor expansion according to the type of [gH−p]-differentiability. This method is primarily proposed for the solution of the non-homogeneous Wave equation with triangular initial boundaries. Several examples are presented to demonstrate the model's theoretical aspect and how it works for various types of [gH−p]-differentiability. According to the results, the method gives an accurate solution to the fuzzy Wave equation.

Far-field analytical solutions of the non-homogeneous Helmholtz and wave equations for spatially non-localized sources.

Non-localized impulsive sources are ubiquitous in underwater acoustic applications. However, analytical expressions of their acoustic field are usually not available. In this work, far-field analytical solutions of the non-homogeneous scalar Helmholtz and wave equations are developed for a class of spatially extended impulsive sources. The derived expressions can serve as benchmarks to verify the accuracy of numerical solvers.

Emission and propagation of sound waves in porous media with active inner heat sources

Wave propagation through porous media with active inner heat sources is investigated in this paper. Through the use of the two-scale asymptotic method of homogenisation, it is found that a macroscopic non-homogeneous wave equation describes the emission and propagation of sound waves in such active porous media. The upscaled model is validated numerically for the cases of single and double porosity media, and shows that the general properties of the effective parameters of the porous media are not altered by the inner heat sources. Instead, the inner heat sources contribute to the non-homogeneous term in the upscaled wave equation. The paper also explores the potential of using active inner heat sources for controlling sound waves incident to the porous media in practical scenarios.

Global space-time Trefftz DG schemes for the time-dependent linear wave equation

In this paper we are concerned with Trefftz discretizations of the time-dependent linear wave equation in anisotropic media in arbitrary space dimensional domains $\Omega \subset \mathbb{R}^d~ (d\in \mathbb{N})$. We propose two variants of the Trefftz DG method, define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations, and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth $h$ and the condition number of the coefficient matrices, respectively. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous wave equation in anisotropic media. In particular, the error analysis holds for the (nonhomogeneous) Dirichlet, Neumann, and mixed boundary conditions from the original PDEs. Furthermore, a strategy to discretize the model in heterogeneous media is proposed. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

Solutions for non-homogeneous wave equations subject to unusual and Neumann boundary conditions

This paper examines the accuracy, stability, and convergence of the forward and inverse solutions for non-homogeneous wave equations in different cases. The first case is when the wave governing equation is subject to the so-called unusual (non-local/non-classical) boundary condition. Where there is a non-local displacement and flux tension boundary data are placed at the very left end of the examined domain (string or rob), combined with Dirichlet and Neumann data at blue the left end of the domain. We explore the ideal value for the rational stiffness coefficient to be used for the second case which concerns moving Dirichlet data to the right and adding over-determination Neumann boundary conditions to the right or left. The forward solutions for the given problems, represented in the flux tension values, are investigated using the finite difference method (FDM) combined with the separation variable method. We also test the effect of the rotational stiffness coefficient on the effectiveness and convergence of those direct solutions. On the other hand, to obtain stable numerical results for the corresponding inverse problems (concerning retrieval of the force source function), the zeroth, the first, and the second-order of the Tikhonov regularization parameters are tested and the minimum error norm is calculated accordingly to find the best estimate for that space-dependent force source. Because such inverse problems are ill-posed, the existence and uniqueness of their solutions are tolerated and verified based on existing theorems in literature. This work includes numerical examples which illustrate the usefulness of the approach for solving non-homogeneous wave equations under novel kinds of boundary conditions.

Inverse Solution of Thermoacoustic Wave Equation for Cylindrical Layered Media

Thermoacoustic imaging is a crossbred approach taking advantages of electromagnetic and ultrasound disciplines, together. A significant number of current medical imaging strategies are based on reconstruction of source distribution from information collected by sensors over a surface covering the region to be imaged. Reconstruction in thermoacoustic imaging depends on the inverse solution of thermoacoustic wave equation. Homogeneous assumption of tissue to be imaged results in degradation of image quality. In our paper, inverse solution of the thermoacoustic wave equation using layered tissue model consisting of concentric annular layers on a cylindrical cross-section is investigated for cross-sectional thermoacustic imaging of breast and brain. By using Green’s functions and surface integral methods we derive an exact analytic inverse solution of thermoacoustic wave equation in frequency domain. Our inverse solution is an extension of conventional solution to layered cylindrical structures. By carrying out simulations, using numerical test phantoms consisting of thermoacoustic sources distributed in the layered model, our layered medium assumption solution was tested and benchmarked with conventional solutions based on homogeneous medium assumption in frequency domain. In thermoacoustic image reconstruction, where the medium is assumed as homogeneous medium, the solution of nonhomogeneous thermoacoustic wave equation results in geometrical distortions, artifacts and reduced image resolution due to inconvenient medium assumptions.

ON COMPACT 4TH ORDER FINITE-DIFFERENCE SCHEMES FOR THE WAVE EQUATION

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional nonhomogeneous wave equation, n≥ 1. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for n≥ 2. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schro¨dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for n≥ 2. For n = 1 and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.

Non-homogeneous wave equation on a cone

ABSTRACT The wave equation in the cone is shown to have a unique solution if u and its partial derivatives in x are in on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the cone, which can be combined with a solution to the homogeneous wave equation in the cone to obtain the full solution.

Well-posedness results for the wave equation generated by the Bessel operator

In this paper, we consider the non-homogeneous wave equation generated by the Bessel operator. We prove the existence and uniqueness of the solution of the non-homogeneous wave equation generated by the Bessel operator. The representation of the solution is given. We obtained a priori estimates in Sobolev type space. This problem was firstly considered in the work of M. Assal [1] in the setting of Bessel-Kingman hypergroups. The technique used in [1] is based on the convolution theorem and related estimates. Here, we use a different approach. We study the problem from the point of the Sobolev spaces. Namely, the conventional Hankel transform and Parseval formula are widely applied by taking into account that between the Hankel transformation and the Bessel differential operator there is a commutation formula [2].

Analytical fuzzy triangular solutions of the wave equation

The analytical fuzzy triangular solutions for both one-dimensional homogeneous and non-homogeneous wave equations with emphasis on the type of [gH-p]-differentiability of solutions are obtained by using the fuzzy D’Alembert’s formulas. In the current article, the existence and uniqueness of the solutions of the homogeneous and non-homogeneous fuzzy wave equation by considering the type of [gH-p]-differentiability of solutions are provided. In a special case, the fuzzy mathematical model of a vibrating string with a fixed end is investigated. Eventually, given to the various examples represented, the efficacy and accuracy of the method are examined.

Application of the Adomian Decomposition Method to the Solution of the Linear Nonhomogeneous One-Dimensional Wave Equation

Application of the Adomian Decomposition Method to the Solution of the Linear Nonhomogeneous One-Dimensional Wave Equation

Overcoming the near boundary error in the solution of non-homogeneous wave equation by a boundary consistent method

In the paper, a novel boundary consistent method, modified from the Fourier sine series method, is proposed to solve a non-homogeneous wave equation with non-homogeneous boundary conditions. Besides the usual Fourier sine series, we supplement two extra terms in the solution to consider the consistency of the wave equation at the boundaries. We point out a “mistake” in the conventional Fourier sine series method without considering the boundary consistency conditions, which might cause large boundary errors of the Fourier sine series solution. Numerical examples confirm the improvement of the accuracy of the boundary consistent method, which not only overcomes the boundary errors but also improves the accuracy in the whole domain about four orders, upon comparing with the conventional Fourier sine series method.

Application of Polylogarithms to the Approximate Solution of the Inhomogeneous Telegraph Equation for the Distortionless Line

The paper deals with a mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortion. This problem is reduced to the analogous one for the one-dimensional inhomogeneous wave equation. Its solution can be found as the sum of the solution for a mixed homogeneous boundary value problem for the corresponding homogeneous wave equation and for the solution of a non-homogeneous wave equation with homogeneous boundary data and zero initial conditions. Solutions to both problems can be found by separating the variables in the form of a series of trigonometric functions of the line point with time-dependent coefficients. Such solutions are inconvenient for real application because they require calculation of a large number of integrals, and it is difficult to estimate the miscalculation. An alternative method for solving this problem is proposed, based on the use of special functions, viz. polylogarithms, which are complex power-series with power coefficients converging in a unit circle. The exact solution of the problem is expressed in the integral form via the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one is expressed in the form of a finite sum via the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All these parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding powers on the segment of the length equal to the period. This makes it possible to effectively find an approximate solution to the problem. Also, a simple and convenient error estimate of the approximate solution of the problem is found. It is linear with respect to the step of splitting the line and the step of splitting the time range in which the problem is considered. The score is uniform along the length of the line at each fixed point of time. A concrete example of solving the problem according to the proposed mode is presented; graphs of exact and approximate solutions are constructed.