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A new solution to the mono-dimensional diffusion equation for time-variable first kind boundary condition is presented where the time-variable function at the surface is derived proposing a surface saturation model. This solution may be…
A new transform method for solving boundary value problems for linear and integrable nonlinear PDEs recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation…
We derive an explicit representation of the fundamental solution to the heat equation in a half-space of ${\mathbb R}^N$ with a diffusive dynamical boundary condition, and establish sharp pointwise upper and lower bounds. We also…
Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain…
Accurate simulation of wave propagation in complex acoustic materials is crucial for applications in sound design, noise control, and material engineering. Traditional numerical solvers, such as finite element methods, are computationally…
In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity…
The inversion theorem and convolution theorem of the conformable fractional Laplace transforms are developed. All the elementary properties of the classical Laplace transform are extended to the conformable fractional transform, and using…
The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the…
We consider a solution $u(\cdot,t)$ to an initial boundary value problem for time-fractional diffusion-wave equation with the order $\alpha \in (0,2) \setminus \{ 1\}$ where $t$ is a time variable. We first prove that a suitable norm of…
In this paper we propose a {\it discontinuous} plane wave neural network (DPWNN) method with $hp-$refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional…
We consider the application of the WaveHoltz iteration to time-harmonic elastic wave equations with energy conserving boundary conditions. The original WaveHoltz iteration for acoustic Helmholtz problems is a fixed-point iteration that…
We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract…
Wavelet theory has been well studied in recent decades. Due to their appealing features such as sparse multiscale representation and fast algorithms, wavelets have enjoyed many tremendous successes in the areas of signal/image processing…
This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, 1) A multiple scattering strategy that decomposes a…
The method of fundamental solutions (MFS), also known as the method of auxiliary sources (MAS), is a well-known computational method for the solution of boundary-value problems. The final solution ("MAS solution") is obtained once we have…
We present a fast direct solver for the volume scattering problem of the Helmholtz equation. The algorithm is faster than existing methods. Moreover, discretization for our method is much simpler and more accurate than that for finite…
We revisit the problem of solving the one-dimensional wave equation on a domain with moving boundary. In J. Math. Phys. 11, 2679 (1970), Moore introduced an interesting method to do so. As only in rare cases, a closed analytical solution is…
In this study, we consider the numerical solution of the Neumann initial boundary value problem for the wave equation in 2D domains. Employing the Laguerre transform with respect to the temporal variable, we effectively transform this…
We consider the well-posedness of the initial-boundary value problem for a time-fractional partial differential equation with the fractional order lying in (1,2]. For the case of time-dependent coefficients, it is difficult to give an…
Inferring electromagnetic propagation characteristics within the marine atmospheric boundary layer (MABL) from data in real time is crucial for modern maritime navigation and communications. The propagation of electromagnetic waves is well…