Related papers: Numerical Solution Partial Differential Equations …
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
The special unitary group SU(2) plays a fundamental role in the description of symmetries in quantum mechanics, theoretical physics, and spherical signal processing. In this paper, we address the computational challenges of performing…
The accuracy of the numerical solution of a fractional differential equation depends on the differentiability class of the solution. The derivatives of the solutions of fractional differential equations often have a singularity at the…
Improving the generalization ability of Deep Neural Networks (DNNs) is critical for their practical uses, which has been a longstanding challenge. Some theoretical studies have uncovered that DNNs have preferences for some frequency…
Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this…
Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to…
The solution of a nonlinear diffusion equation is numerically investigated using the generalized Fourier transform method. This equation includes fractal dimensions and power-law dependence on the radial variable and on the diffusion…
The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…
Fractional calculus has been used to describe physical systems with complexity. Here, we show that a fractional calculus approach can restore or include complexity in any physical systems that can be described by partial differential…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of…
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in…
We present four quantum algorithms for solving a multidimensional drift-diffusion equation. They rely on a quantum linear system solver, a quantum Hamiltonian simulation, a quantum random walk, and the quantum Fourier transform. We compare…
In this paper, we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions by means of spectral methods. The main features of this method are its…
The convolution potential arises in a wide variety of application areas, and its efficient and accurate evaluation encounters three challenges: singularity, nonlocality and anisotropy. We introduce a fast algorithm based on a far-field…
Fractional wave equation arises in different type of physical problems such as the vibrating strings, propagation of electro-magnetic waves, and for many other systems. The exact analytical solution of the fractional differential equation…
We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…
Solving partial differential equations is difficult. Recently proposed neural resolution-invariant models, despite their effectiveness and efficiency, usually require equispaced spatial points of data. However, sampling in spatial domain is…
This paper investigates, a new class of fractional order Runge-Kutta (FORK) methods for numerical approximation to the solution of fractional differential equations (FDEs). By using the Caputo generalizedTaylor formula and the total…