# On a numerical method for a homogeneous, nonlinear, nonlocal, elliptic boundary value problem

John R. Cannon, Daniel J. Galiffa

Research output: Contribution to journalArticle

8 Citations (Scopus)

### Abstract

In this work we develop a numerical method for the equation: -α(∫01u(t)dt)u″(x)+[u(x)]2n+1=0,x∈(0,1),u(0)=a,u(1)=b. We begin by establishing a priori estimates and the existence and uniqueness of the solution to the nonlinear auxiliary problem via the Schauder fixed point theorem. From this analysis, we then prove the existence and uniqueness to the problem above by defining a continuous compact mapping, utilizing the a priori estimates and the Brouwer fixed point theorem. Next, we analyze a discretization of the above problem and show that a solution to the nonlinear difference problem exists and is unique and that the numerical procedure converges with error (h). We conclude with some examples of the numerical process.

Original language English (US) 1702-1713 12 Nonlinear Analysis, Theory, Methods and Applications 74 5 https://doi.org/10.1016/j.na.2010.10.042 Published - Mar 1 2011

### Fingerprint

Elliptic Boundary Value Problems
Boundary value problems
Numerical methods
Numerical Methods
A Priori Estimates
Existence and Uniqueness
Brouwer Fixed Point Theorem
Schauder Fixed Point Theorem
Numerical Procedure
Discretization
Converge

### All Science Journal Classification (ASJC) codes

• Analysis
• Applied Mathematics

### Cite this

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In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 74, No. 5, 01.03.2011, p. 1702-1713.

Research output: Contribution to journalArticle

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T1 - On a numerical method for a homogeneous, nonlinear, nonlocal, elliptic boundary value problem

AU - Cannon, John R.

AU - Galiffa, Daniel J.

PY - 2011/3/1

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N2 - In this work we develop a numerical method for the equation: -α(∫01u(t)dt)u″(x)+[u(x)]2n+1=0,x∈(0,1),u(0)=a,u(1)=b. We begin by establishing a priori estimates and the existence and uniqueness of the solution to the nonlinear auxiliary problem via the Schauder fixed point theorem. From this analysis, we then prove the existence and uniqueness to the problem above by defining a continuous compact mapping, utilizing the a priori estimates and the Brouwer fixed point theorem. Next, we analyze a discretization of the above problem and show that a solution to the nonlinear difference problem exists and is unique and that the numerical procedure converges with error (h). We conclude with some examples of the numerical process.

AB - In this work we develop a numerical method for the equation: -α(∫01u(t)dt)u″(x)+[u(x)]2n+1=0,x∈(0,1),u(0)=a,u(1)=b. We begin by establishing a priori estimates and the existence and uniqueness of the solution to the nonlinear auxiliary problem via the Schauder fixed point theorem. From this analysis, we then prove the existence and uniqueness to the problem above by defining a continuous compact mapping, utilizing the a priori estimates and the Brouwer fixed point theorem. Next, we analyze a discretization of the above problem and show that a solution to the nonlinear difference problem exists and is unique and that the numerical procedure converges with error (h). We conclude with some examples of the numerical process.

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