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Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on
Hector Vazquez-Leal1, Brahim Benhammouda2, Uriel Antonio Filobello-Nino1
1Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Mexico.
Springerplus
|May 3, 2014
Summary
A new modified Taylor series method (MTSM) effectively approximates nonlinear problems. This approach overcomes limitations of standard Taylor series methods for boundary value problems, yielding accurate and computable results.
Area of Science:
- Applied Mathematics
- Numerical Analysis
- Differential Equations
Background:
- Nonlinear problems on finite intervals pose significant challenges for traditional approximation methods.
- Existing Taylor series methods struggle with mixed boundary conditions in nonlinear problems.
Purpose of the Study:
- To introduce a modified Taylor series method (MTSM) for approximating solutions to nonlinear problems on finite intervals.
- To address the limitations of standard Taylor series methods when dealing with mixed boundary conditions.
Main Methods:
- The modified Taylor series method (MTSM) is proposed, utilizing shooting constants and extra derivatives.
- The method is applied to solve three distinct types of nonlinear boundary value problems (BVPs).
Main Results:
- The MTSM successfully approximates solutions for third-order BVPs with hyperbolic sine nonlinearity.
- Accurate approximations were achieved for second-order nonlinear differential equations with exponential nonlinearity.
- The method demonstrated efficacy for third-order nonlinear BVPs involving radical nonlinearity.
Conclusions:
- The modified Taylor series method (MTSM) provides a robust and efficient approach for nonlinear problems.
- MTSM generates easily computable and highly accurate approximations for a range of nonlinear equations.
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