On Error Estimation and Convergence of the Difference Scheme for a Nonlinear Elliptic Equation with an Integral Boundary Condition

Second order singularly perturbed delay differential equations with non-local boundary condition

In this paper, we discuss by means of a fixed point theorem, the existence of positive solutions of a system of nonlinear Caputo fractional differential equations with integral boundary conditions. An example is given to illustrate the main results.

On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions

In this paper, we state and prove theorems related to the existence and multiplicity for positive solutions of a system of first order differential equations with multipoint and integral boundary conditions. The main tool is the fixed point theory. In order to illustrate the main results, we present some examples.

A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions

Partial differential equations with nonlocal boundary conditions have been widely applied in various fields of science and engineering. In this work, we first build a high accuracy difference scheme for Poisson equation with two integral boundary conditions. Then, we prove that the scheme can reach the asymptotic optimal error estimate in the maximum norm through applying the discrete Fourier transformation. In the end, numerical experiments validate the correctness of theoretical results and show the stability of the scheme.

An iterative algorithm is proposed for solving the solution of a nonlinear fourth-order differential equation with integral boundary conditions. Its approximate solutionun(x)is represented in the reproducing kernel space. It is proved thatun(x)converges uniformly to the exact solutionu(x). Moreover, the derivatives ofun(x)are also convergent to the derivatives ofu(x). Numerical results show that the method employed in the paper is valid.

We propose an Adomian decomposition method to solve a class of nonlinear differential equations of fractional‐order with modified Caputo derivatives and integral boundary conditions. Our approach uses the integral boundary conditions to derive an equivalent nonlinear Volterra integral equation before establishing existence and uniqueness of solution and a recursion scheme for the solution. The convergence of the method is proved and an error analysis given. Two numerical examples are solved by obtaining a rapidly converging sequence of analytical functions to the solution.

The uniqueness of positive solution for a class of singular fractional differential system with integral boundary conditions is considered in this paper and many types of equation system are contained in this equation system because there are many parameters which can be changeable in this equation system. The fractional orders are involved in the nonlinearity of the boundary value problem and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable. The existence of uniqueness of positive solution is mainly obtained by fixed point theorem of mixed monotone operator and the positive solution of equation system is dependent on λ. An iterative sequence and convergence rate are given which are important for practical application and an example is given to demonstrate the validity of our main results.

We report a new 9-point compact discretization of order two in y- and order four in x-directions, based on cubic spline approximation, for the solution of two-dimensional nonlinear elliptic partial differential equations of the form $$ A\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial x^{2} }} + B\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial y^{2} }} = f\left( {x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}} \right),\left( {x,y} \right) \in\Omega $$ defined in the domain \( \Omega = \left\{ {\left( {x,y} \right):0 0 \) and \( B\left( {x,y} \right) > 0 \) in \( \Omega \). The corresponding Dirichlet boundary conditions are prescribed by $$ u\left( {x,y} \right) = \psi \left( {x,y} \right),\quad \left( {x,y} \right) \in \partial\Omega $$ The main spline relations are presented and incorporated into solution procedures for elliptic partial differential equations. Available numerical methods based on cubic spline approximations for the numerical solution of nonlinear elliptic equations are of second-order accurate. Although 9-point finite difference approximations of order four accurate for the solution of nonlinear elliptic differential equations are discussed in the past, but these methods require five evaluations of the function f. In this piece of work, using the same number of grid points and three evaluations of the function f, we have derived a new stable cubic spline method of order 2 in y- and order 4 in x-directions for the solution of nonlinear elliptic equation. However, for a fixed parameter (Δy/Δx 2), the proposed method behaves like a fourth order method. The accuracy of the proposed method is exhibited from the computed results. The proposed method is applicable to Poisson’s equation and two-dimensional Navier-Stokes’ equations of motion in polar coordinates, which is main highlight of the work. The convergence analysis of the proposed cubic spline approximation for the nonlinear elliptic equation is discussed and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.

By employing known Guo-Krasnoselskii fixed point theorem, we investigate the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional differential equation with integral boundary conditions.

In this paper the existence and uniqueness of solutions to the fractional differential equations with two-point and integral boundary conditions is investigated. The Green function is constructed, and the problem under consideration is reduced to the equivalent integral equation. Existence and uniqueness of a solution to this problem is analyzed using the Banach the contraction mapping principle and Krasnoselskii’s fixed point theorem.

The present paper is devoted to study a nonlinear second order differential equation with iterative source term and integral boundary conditions. The utilization of some suitable fixed point theorems ultimately led us to establish some sufficient conditions that guarantee the existence, uniqueness and continuous dependence of positive bounded solutions. The obtained results are illustrated by an example.

Viscoelastic Kelvin–Voigt model on Ulam–Hyer’s stability and T-controllability for a coupled integro fractional stochastic systems with integral boundary conditions via integral contractors

In this paper, based on the upper and lower solutions method and monotone iterative techniques, we study the existence of solutions for fractional differential systems with continuous nonlinearities and integral boundary conditions.The construction of the monotone sequences and the definition of upper and lower solutions depend on the quasimonotone property of the reaction functions.We prove the existence of maximal and minimal solutions for quasimonotone increasing systems.Also, we prove the existence of maximal-minimal and minimal-maximal solutions for quasimonotone decreasing systems and for mixed quasimonotone systems and the existence of at least one solution.Finally, we give some examples to illustrate our results.