Related papers: A new modified highly accurate Laplace-Fourier met…
We study approximations to the Moreau envelope -- and infimal convolutions more broadly -- based on Laplace's method, a classical tool in analysis which ties certain integrals to suprema of their integrands. We believe the connection…
The leading asymptotics of the truncation error for Gauss's continued fraction is determined exactly. Not only for this purpose but also for wider applicability elsewhere the discrete analogue of Laplace's method for hypergeometric series…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…
We consider difference equations of the form $x_{n+1}=F_0(x_n,\ldots,x_{n-k+1}),$ and increase the delay through a process of successive substitutions to obtain a sequence of systems $y_{n+1}=F_j(x_{n-j},\ldots,x_{n-k-j+1}),\;…
In this work, we analyse the discretisation of a recently proposed new Lagrangian approach to optimal control problems of affine-controlled second-order differential equations with cost functions quadratic in the controls. We propose exact…
We introduce an exact parameterized extended system such that, under adequate data, between the components of its solution, there is the solution of the weak formulation of the homogeneous Dirichlet problem for the stationary Stokes…
Various computational challenges arise when applying Bayesian inference approaches to complex hierarchical models. Sampling-based inference methods, such as Markov Chain Monte Carlo strategies, are renowned for providing accurate results…
We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First,…
This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented…
It is well known that Newton's method can have trouble converging if the initial guess is too far from the solution. Such a problem particularly occurs when this method is used to solve nonlinear elliptic partial differential equations…
In this paper, approximate solutions for a class of fractional Lane - Emden type equations based on the series expansion method are presented. Various examples are introduced and discussed. The recurrence relation for the components of the…
Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be non-robust in the presence of Neumann boundary conditions. In this paper we overcome this issue by formulating the RBF-generated finite…
The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle…
This paper focuses on the study of integro-differential equations with delays, presenting a novel perturbation approach. The primary objective is to introduce the concepts of classical and mild solutions for these equations and establish…
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local…
We consider the asymptotic behavior of the multidimensional Laplace-type integral with a perturbed phase function. Under suitable assumptions, we derive a higher-order asymptotic expansion with an error estimate, generalizing some previous…
A new analytic approximate technique for addressing nonlinear problems, namely the optimal perturbation iteration method, is introduced and implemented to singular initial value Lane-Emden type problems to test the effectiveness and…
Deep learning method is of great importance in solving partial differential equations. In this paper, inspired by the failure-informed idea proposed by Gao et.al. (SIAM Journal on Scientific Computing 45(4)(2023)) and as an improvement, a…
The present work concerns a version of the Fisher-KPP equation where the nonlinear term is replaced by a saturation mechanism, yielding a free boundary problem with mixed conditions. Following an idea proposed in [BrunetDerrida.2015], we…