Nonhomogeneous Wave Equation Research Articles

Fractional electromagnetic waves in circular waveguides with fractional-order inductance characteristics

ABSTRACTThe Maxwell equations play a vital role in the electromagnetic theory. With the development of Fractional Calculus and its applications, new fractional Maxwell equations are required to describe fractional electromagnetic fields. In this paper, the fractional electromagnetic induction law is presented with considering the properties of fractional inductance. Based on this, we have obtained the fractional Maxwell equations. In addition, the non-homogeneous fractional wave equations and the homogeneous fractional vector Helmholtz equations are deduced. Using these results, we investigate the behavior of electromagnetic waves in fractional circular waveguides with fractional-order inductance characteristics. The results show that the electromagnetic fields change with different fractional orders.

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

Classical and Generalized Solutions of a Mixed Problem for a Nonhomogeneous Wave Equation

A.N. Krylov’s ideas concerning convergence acceleration for Fourier series are used to obtain explicit expressions for the classical solution of a mixed problem for a nonhomogeneous equation and explicit expressions for a generalized solution in the case of arbitrary summable $$q(x)$$ , $$\varphi (x)$$ , $$\psi (x)$$ , and $$f(x,t)$$ .

Classical and Generalized Solutions of a Mixed Problem for a Nonhomogeneous Wave Equation

A.N. Krylov’s ideas concerning convergence acceleration for Fourier series are used to obtain explicit expressions for the classical solution of a mixed problem for a nonhomogeneous equation and explicit expressions for a weak solution in the case of arbitrary summable $$q(x)$$ , $$\varphi (x)$$ , $$\psi (x)$$ , and $$f(x,t)$$ .

Method of Riesz potentials applied to solution to nonhomogeneous singular wave equations

Riesz potentials are convolution operators with fractional powers of some distance (Euclidean, Lorentz or other) to a point. From application point of view, such potentials are tools for solving differential equations of mathematical physics and inverse problems. For example, Marsel Riesz used these operators for writing the solution to the Cauchy problem for the wave equation and theory of the Radon transform is based on Riesz potentials. In this article, we use the Riesz potentials constructed with the help of generalized convolution for solution to the wave equations with Bessel operators. First, we describe general method of Riesz potentials, give basic definitions, introduce solvable equations and write suitable potentials (Riesz hyperbolic $B$-potentials). Then, we show that these potentials are absolutely convergent integrals for some functions and for some values of the parameter representing fractional powers of the Lorentz distance. Next we show the connection of the Riesz hyperbolic B-potentials with d’Alembert operators in which the Bessel operators are used in place of the second derivatives. Next we continue analytically considered potentials to the required parameter values that includes zero and show that when value of the parameter is zero these operators are identity operators. Finally, we solve singular initial value hyperbolic problems and give examples.

The retarded potential of a non-homogeneous wave equation: introductory analysis through the Green Functions

The retarded potential, a solution of the non-homogeneous wave equation, is a subject of particular interest in many physics and engineering applications. Examples of such applications may be the problem of solving the wave equation involved in the emission and reception of a signal in a synthetic aperture radar (SAR), scattering and backscattering, and general electrodynamics for media free of magnetic charges.However, the construction of this potential solution is based on the theory of distributions, a topic that requires special care and time to be understood with mathematical rigor. Thus, the goal of this study is to provide an introductory analysis, with a medium level of formalism, on the construction of this potential solution and the handling of Green functions represented by sequences of well-behaved approximatingfunctions.

Two Very Accurate and Efficient Methods for Solving Time-Dependent Problems

In this paper, collocation method based on Bernoulli and Galerkin method based on wavelet are proposed for solving nonhomogeneous heat and wave equations. The two methods have the linear systems solved by suitable solvers. Several examples are given to examine the performance of these methods and a comparison is made.

A new method to solve non-homogeneous wave equations of electromagnetic fields by fourier’s triple integral transform

This paper uses Fourier’s triple integral transform method to simplify the calculation of the non-homogeneous wave equations of the time-varying electromagnetic field. By adding several special definite conditions to the wave equation, it becomes a mathematical problem of definite condition. Then by using Fourier’s triple integral transform method, this three-dimension non-homogeneous partial differential wave equation is changed into an ordinary differential equation. Through the solution to this ordinary differential equation, the expression of the relationship between the time-varying scalar potential and electromagnetic wave excitation source is developed precisely. This method simplifies the solving process effectively.

Error analysis of sinc-Galerkin method for time-dependent partial differential equations

In this paper, we apply the Galerkin method with sinc bases for solving the boundary-value problems involving nonhomogeneous heat, convection diffusion, wave, and telegraph equations. The accuracy of the method for equation is $ \mathrm {O} (({\Delta } t)+e^{-k\sqrt {N} })$ . Error analysis is included. Several examples are given to illustrate the efficiency and implementation of the sinc-Galerkin method. Comparisons are made to confirm the reliability of the method.

Simulation Model of Propagation of Vibrations in a System of Connected Bodies for the Solution of Problems of Vibration Diagnostics

An approach to the construction of a mathematical model of the process of propagation of acoustic vibrations in equipment that is in the form of a system of connected bodies is considered. The proposed structure of the mathematical model is implemented as a graph whose edges are links between the elements of a system of connected bodies formalized by nonhomogeneous wave equations that take into account dissipative losses. Through the use of the mathematical model together with the introduction of sources of an external disturbance in any node of the graph, it becomes possible to describe a variation in the vibration signal at a contact point, estimate its properties, and select on a well-defined basis the point of attachment of the vibration signal sensor. The mathematical model is tested to assure it corresponds qualitatively to the actual processes in the propagation of vibration signals by means of test disturbances, and it is shown that it may be used to evaluate the selection of the points of attachment of the vibration signal sensors in problems of vibration diagnostics of equipment.

РЕЗОЛЬВЕНТНЫЙ ПОДХОД К МЕТОДУ ФУРЬЕ В СМЕШАННОЙ ЗАДАЧЕ ДЛЯ НЕОДНОРОДНОГО ВОЛНОВОГО УРАВНЕНИЯ

РЕЗОЛЬВЕНТНЫЙ ПОДХОД К МЕТОДУ ФУРЬЕ В СМЕШАННОЙ ЗАДАЧЕ ДЛЯ НЕОДНОРОДНОГО ВОЛНОВОГО УРАВНЕНИЯ

Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods

In this article, a modified cubic B-spline differential quadrature method (MCB-DQM) is proposed to solve a hyperbolic diffusion problem in which flow motion is affected by both convection and diffusion. One dimensional hyperbolic non-homogeneous heat, wave and telegraph equations are also considered along with two dimensional hyperbolic diffusion problem. The method reduces the hyperbolic problem into a system of nonlinear ordinary differential equations. The system is then solved by the optimal four stage three order strong stability-preserving time stepping Runge–Kutta (SSP-RK43) scheme. The reliability and efficiency of the method has been tested on seven examples. The stability of the method is also discussed and found to be unconditionally stable.

Neumann-Boundary Stabilization of the Wave Equation with Internal Damping Control and Applications

This paper is devoted to the Neumann boundary stabilization of a non-homogeneous ndimensional wave equation subject to static or dynamic boundary conditions. Using a linear feedback law involving only an internal term, we prove the well-posedness of the considered systems and provide a simple method to obtain an asymptotic convergence result for the solutions. The method consists of proposing a new energy norm, and applying the semigroup theory and LaSalle's principle. Finally, the method presented in this work is also applied to several distributed parameter systems such as the Petrovsky system, coupled wave-wave equations and elastic system.

Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertexOof the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance toO. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

Singular solutions with exponential growth to Protter’s problems

Four-dimensional boundary value problems which were formulated by Proter for the nonhomogeneous wave equation are studied. They can be considered as multidimensional versions of the Darboux problems in ℝ2. Protter’s problem is not well posed in the frame of classical solvability. On the other hand, it is known that the unique generalized solution may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Some known results suggest that the solution may have at most exponential growth. We construct an infinitely smooth right-hand side function such that the corresponding generalized solution to Protter’s problem has an exponential singularity.

Protter-Morawetz multidimensional problems

About 50 years ago M.H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter-Morawetz multidimensional problems, and an understanding of the situation is not at hand. At the same time, Protter also formulated boundary value problems in the hyperbolic part of the domain—the nonhomogeneous wave equation is studied in a (3+1)-D domain bounded by two characteristic cones and a non-characteristic ball. These problems could be considered as multidimensional variants of the Darboux problem in ℝ2. In the frame of classical solvability the hyperbolic Protter problem is not Fredholm, because it has an infinite-dimensional cokernel. On the other hand, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity even for some very smooth right-hand side functions. This singularity is isolated at the vertex O of the boundary light cone and does not propagate along the characteristic cone. In the general case of smooth right-hand side function, some necessary and sufficient conditions for the existence of a bounded solution are given and a priori estimates for the solution are found. The semi-Fredholm solvability of the problem is proved.

An integral equation approach to acoustic cloaking

The paper presents an original boundary integral equation (BIE) formulation for the analysis of the acoustic cloaking of a scatterer. The advantage of such an approach is the lower computational burden, especially when the analysis of a large portion of the hosting domain is required. The partial differential equation governing the propagation inside the cloak is recast in the form of non-homogeneous wave equation, with field sources depending on the mechanical properties of the cloak. The boundary integral formulation is derived using the standard procedure. The boundary element method (BEM) is used to derive the matrix transfer function of the cloak. The latter is applied to the incident field at the cloak's outer boundary to obtain the total field at arbitrary locations in the host. The formulation is applied to the simple case of a radially symmetric cloak embedding a circular obstacle. Numerical results are presented for sound-hard and sound-soft obstacles, including a study of the cloaking efficiency as a function of the frequency.

Thermoacoustic tomography in elastic media

We investigate the problem of recovering the initial displacement f for a solution u of a linear, isotropic, non-homogeneous elastic wave equation, given measurements of u on [0, T] × ∂Ω, where is some bounded domain containing the support of f. For the acoustic wave equation, this problem is known as thermoacoustic tomography, and has been well studied; for the elastic wave equation, matters are complicated by the presence of two speeds of propagation, and we give sufficient conditions on the Lamé parameters to ensure that recovery is possible.

Solution to time fractional partial differential equations via joint Laplace-Fourier transforms

In this article, the authors solved certain non-homogeneous time fractional wave equations and dynamical systems with fractional derivatives using integral transforms. Transform method is a powerful tool for solving partial fractional differential equations. The result reveals that the transform method is very convenient and effective.

Nonlinear diffraction of super-gaussian laser beams in an array of quantum-dot chains with coulomb interaction taken into account

We consider the propagation of super-Gaussian monochromatic laser beams in a three-dimensional array of quantum dots coupled by the tunneling effect along one axis. The electron energy spectrum of the system corresponds to the Hubbard model, where the Coulomb interaction of electrons in quantum dots is taken into account. The field of the laser beam is described by the Maxwell equations, from which a nonhomogeneous wave equation for the vector potential is obtained. In the approximation of slowly varying amplitudes and phases, the wave equation is reduced to a phenomenological equation describing the electromagnetic field in an array of chains of quantum dots. We study the influence of the system parameters and the frequency of the laser-beam field on the propagation in the medium by solving numerically the phenomenological equation. We obtain the dependence of the factor characterizing the diffraction blooming of the beam in an array of chains of quantum dots on the parameters of the system’s electron energy spectrum.