Related papers: On the Perturbation-Iteration Algorithm for System…
Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on…
We consider quadrature formulas of high order in time based on Radau-type, L-stable implicit Runge-Kutta schemes to solve time dependent stiff PDEs. Instead of solving a large nonlinear system of equations, we develop a method that performs…
We propose algorithms for solving high-dimensional Partial Differential Equations (PDEs) that combine a probabilistic interpretation of PDEs, through Feynman-Kac representation, with sparse interpolation. Monte-Carlo methods and…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
We propose an efficient algorithm for the approximation of fractional integrals by using Runge--Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
A robust and fast solver for the fractional differential equation (FDEs) involving the Riesz fractional derivative is developed using an adaptive finite element method on non-uniform meshes. It is based on the utilization of hierarchical…
In this paper, we introduce a novel approach called the Iterative Aboodh Transform Method (IATM) which utilizes Daftardar--Jafari polynomials for solving non-linear problems. Such method is employed to derive solutions for non-linear…
An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and…
We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure…
In this work, we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schr\"odinger-type,…
Rational solutions of partial differential equations (PDEs) are notoriously difficult to approximate via spectral Fourier methods due to their algebraically slow decay rate. In this work we discuss approximating rational PDE solutions in a…
One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating…
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…
Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation.…
Fractional-order differential equations (FDEs) enhance traditional differential equations by extending the order of differential operators from integers to real numbers, offering greater flexibility in modeling complex dynamical systems…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class…
In this paper, a non-polynomial spectral Petrov-Galerkin method and associated collocation method for substantial fractional differential equations (FDEs) are proposed, analyzed, and tested. We extend a class of generalized Laguerre…