Vol. 7, No. 5, 2014

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Superlinear convergence via mixed generalized quasilinearization method and generalized monotone method

Vinchencia Anderson, Courtney Bettis, Shala Brown, Jacqkis Davis, Naeem Tull-Walker, Vinodh Chellamuthu and Aghalaya S. Vatsala

Vol. 7 (2014), No. 5, 699–712
Abstract

The method of upper and lower solutions guarantees the interval of existence of nonlinear differential equations with initial conditions. To compute the solution on this interval, we need coupled lower and upper solutions on the interval of existence. We provide both theoretical as well as numerical methods to compute coupled lower and upper solutions by using a superlinear convergence method. Further, we develop monotone sequences which converge uniformly and monotonically, and with superlinear convergence, to the unique solution of the nonlinear problem on this interval. We accelerate the superlinear convergence by means of the Gauss–Seidel method. Numerical examples are developed for the logistic equation. Our method is applicable to more general nonlinear differential equations, including Riccati type differential equations.

Keywords
coupled lower and upper solutions, superlinear convergence
Mathematical Subject Classification 2010
Primary: 34A12
Secondary: 34A34
Milestones
Received: 15 September 2013
Revised: 22 November 2013
Accepted: 24 November 2013
Published: 1 August 2014

Communicated by Johnny Henderson
Authors
Vinchencia Anderson
Medgar Evers College
1650 Bedford Avenue
New York, NY 11225
United States
Courtney Bettis
Xavier University of Louisiana
1 Drexel Drive
New Orleans, LA 70125
United States
Shala Brown
Spelman College
350 Spelman Lane SW
Atlanta, GA 30314
United States
Jacqkis Davis
Department of Mathematics
University of Louisiana at Lafayette
104 East University Avenue
Lafayette, LA 70504
United States
Naeem Tull-Walker
Grambling State University
403 Main Street
Grambling, LA 71245
United States
Vinodh Chellamuthu
Department of Mathematics
University of Louisiana at Lafayette
104 East University Avenue
Lafayette, LA 70504-1010
United States
Aghalaya S. Vatsala
Department of Mathematics
University of Louisiana at Lafayette
104 East University Avenue
Lafayette, LA 70504-1010
United States