### 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) |
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Pages (from-to) | 1702-1713 |

Number of pages | 12 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 74 |

Issue number | 5 |

DOIs | |

State | Published - Mar 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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*Nonlinear Analysis, Theory, Methods and Applications*, vol. 74, no. 5, pp. 1702-1713. https://doi.org/10.1016/j.na.2010.10.042

**On a numerical method for a homogeneous, nonlinear, nonlocal, elliptic boundary value problem.** / Cannon, John R.; Galiffa, Daniel J.

Research output: Contribution to journal › Article

TY - JOUR

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

Y1 - 2011/3/1

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.

UR - http://www.scopus.com/inward/record.url?scp=78651379279&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651379279&partnerID=8YFLogxK

U2 - 10.1016/j.na.2010.10.042

DO - 10.1016/j.na.2010.10.042

M3 - Article

AN - SCOPUS:78651379279

VL - 74

SP - 1702

EP - 1713

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 5

ER -