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We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…
We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional…
We present a new view onto the successive approximations' approach in study of the two-point nonlinear fractional boundary value problems. In order to reduce the original problem and further construct its approximate solution we use the…
To extract the approximate solutions in the case of nonlinear fractional order differential equations with the homogeneous and nonhomogeneous boundary conditions, the weighted residual method is embedded here. We exploit three methods such…
The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…
Recently, the nonlinearity continuation method has been used to numerically solve boundary value problems for steady-state Richards equation. The method can be considered as a predictor-corrector procedure with the simplest form which has…
In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and…
By topological arguments, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions of a class of perturbed nonlinear integral equations. These type of integral equations arise, for example,…
In this paper the local order of convergence used in iterative methods to solve nonlinear systems of equations is revisited, where shorter alternative analytic proofs of the order based on developments of multilineal functions are shown.…
This paper presents a novel shooting method for solving two-point boundary value problems for second order ordinary differential equations. The method works as follows: first, a guess for the initial condition is made and an integration of…
We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility…
The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the…
This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary,…
In this paper a special type of difference equations is investigated. The impulses start abruptly at some points and their action continue on given finite intervals. This type of equations is used to model a real process. An algorithm,…
In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results…
We explore singular second-order boundary value problems with mixed boundary conditions on a general time scale. Using the lower and upper solutions method combined with the Brouwer fixed point theorem we demonstrate the existence of a…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
In this work we obtain sufficient conditions for the existence of bounded solutions of a resonant multi-point second-order boundary value problem, with a fully differential equation. The noninvertibility of the linear part is overcome by a…
A new numerical method for solving a scalar ordinary differential equation with a given initial condition is introduced. The method is using a numerical integration procedure for an equivalent integral equation and is called in this paper…
Using direct variational method we consider the existence of non-spurious solutions to the following Dirichlet problem $\ddot{x}\left( t\right) =f\left( t,x\left( t\right) \right) $, $x\left( 0\right) =x\left( 1\right) =0 $ where $f:\left[…