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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
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 34A12
Secondary: 34A34
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Milestones
Received: 15 September 2013
Revised: 22 November 2013
Accepted: 24 November 2013
Published: 1 August 2014
Communicated by Johnny Henderson
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Authors |
| Vinchencia Anderson | |
| Medgar Evers College 1650 Bedford Avenue New York, NY 11225 United States |
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| Courtney Bettis | |
| Xavier University of Louisiana 1 Drexel Drive New Orleans, LA 70125 United States |
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| Shala Brown | |
| Spelman College 350 Spelman Lane SW Atlanta, GA 30314 United States |
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| Jacqkis Davis | |
| Department of Mathematics University of Louisiana at Lafayette 104 East University Avenue Lafayette, LA 70504 United States |
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| Naeem Tull-Walker | |
| Grambling State University 403 Main Street Grambling, LA 71245 United States |
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| Vinodh Chellamuthu | |
| Department of Mathematics University of Louisiana at Lafayette 104 East University Avenue Lafayette, LA 70504-1010 United States |
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| Aghalaya S. Vatsala | |
| Department of Mathematics University of Louisiana at Lafayette 104 East University Avenue Lafayette, LA 70504-1010 United States |
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