An Extension of the Reflected Waves Method to the IBVP for the 1D Non-Homogeneous Wave Equation

This article discusses two significant results about the uniform convergence of Fourier series. One is the uniform convergence of the Fourier series for Hölder-α(α>0) continuous functions, while the other is the uniform convergence of the Fourier series for functions that can be expressed as the sum of finitely many continuous monotonic functions. The two results are particularly useful. Because Hölder continuity condition ensures a specific regularity that facilitates the analysis and provides a robust framework for proving uniform convergence and many practical functions can be decomposed into monotonic components, making the theorem broadly applicable. By providing detailed proofs and applications, this study demonstrates the significant implications of uniform convergence in both theoretical and practical contexts. Moreover, this work paves the way for future research in higher-dimensional Fourier analysis, nonperiodic functions, and adaptive Fourier techniques, underscoring the ongoing relevance and versatility of Fourier series in modern mathematics. Several well-known results emerge as corollaries of these findings, highlighting the robustness and applicability of these theorems in Fourier analysis.

This article discusses two significant results about the uniform convergence of Fourier series. One is the uniform convergence of the Fourier series for Hölder-α(α>0) continuous functions, while the other is the uniform convergence of the Fourier series for functions that can be expressed as the sum of finitely many continuous monotonic functions. The two results are particularly useful. Because Hölder continuity condition ensures a specific regularity that facilitates the analysis and provides a robust framework for proving uniform convergence and many practical functions can be decomposed into monotonic components, making the theorem broadly applicable. By providing detailed proofs and applications, this study demonstrates the significant implications of uniform convergence in both theoretical and practical contexts. Moreover, this work paves the way for future research in higher-dimensional Fourier analysis, non-periodic functions, and adaptive Fourier techniques, underscoring the ongoing relevance and versatility of Fourier series in modern mathematics. Several well-known results emerge as corollaries of these findings, highlighting the robustness and applicability of these theorems in Fourier analysis.

The large time asymptotics of solutions of the Cauchy problem for a non-homogeneous wave equation with a periodic perturbation is found under some restrictions, imposed on the corresponding operator. We apply the obtained results to the third boundary value problem for the wave equation in a half-space.

We study four-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of Darboux problems on the plane. They were formulated by M.H. Protter in connection with BVPs for mixed type equations that model transonic flow phenomena. It is known that the unique generalized solution of Protter's problem may have singularity at only one point. This singularity is isolated at the vertex O of the boundary light characteristic cone and does not propagate along the cone. We present some conditions on the smooth right-hand side functions that are sufficient for existence of generalized solutions and give some a priori estimates for its singularity at O.

Almost‐everywhere singular (AES) distributions, usually referred to as multifractal measures, provide an intermediate link between atomic distributions (distributions represented by a countable superposition of Dirac's delta terms) and smooth regular distributions. This article shows how AES distributions can be rigorously treated in connection with distributed‐parameter models and presents closed‐form expressions and/or recursive, uniformly converging approximation methods for integral transforms (Laplace and Stieltjes). In particular, exact results are obtained and discussed for linear and uniform nonlinear kinetcis and for transport schemes in the presence of continuous mixtures. The physical origin of AES distributions in real systems is also detailed.

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

The article is devoted to equations of the form (*)where is a continuous function of bounded variation on containing a singular component. First we study asymptotic and other properties of the solutions of formal Volterra equations (*) corresponding to for . Next we introduce and study non-linear factorization equations (NFE) for (*). Factorization is constructed in the case when , in , and . With the aid of this factorization, we prove existence theorems for homogeneous and non-homogeneous equations in the singular case .

A model of non-homogeneous damped electromagnetic wave and heat equation in ferrite materials

Global existence for the nonhomogeneous quasilinear wave equation with a localized weakly nonlinear dissipation in exterior domains

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.

This paper, by using Hausdorff calculus theory, decomposes farther a singular continuous distribution function into a sum of a series of absolute continuous distribution functions in different levels with respect to the Hausdorff measures and a singular continuous distribution function with respect to the Hausdorff measures. Consequently, it gives a more accurate decomposition formula for the distribution functions.

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.

We consider a finite element method for the nonhomogeneous second-order wave equation, which is formulated in terms of continuous approximation functions in both space and time, thereby giving a unified treatment of the spatial and temporal discretizations. Our analysis uses primarily energy arguments, which are quite common for spatial discretizations but not for time. We present a priori nodal (in time) superconvergence error estimates without any special time step restrictions. Our method is based on tensor-product spaces for the full discretization.

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.

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.