Spectral method for solving high order nonlinear boundary value problems via operational matrices

2015 ◽  
Vol 55 (4) ◽  
pp. 901-925 ◽  
Author(s):  
Mahmoud Behroozifar
Keyword(s):  
Boundary Value Problems ◽  
Spectral Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
High Order ◽  
Operational Matrices ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value

scholarly journals An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
A. H. Bhrawy ◽  
A. S. Alofi ◽  
R. A. Van Gorder
Keyword(s):  
Boundary Value Problems ◽  
Spectral Method ◽  
Collocation Method ◽  
Nonlinear Boundary ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Mathematical Physics ◽  
Nonlinear Boundary Value Problems ◽  
Polynomial Degree ◽  
Nonlinear Boundary Value

We present a numerical method for a class of boundary value problems on the unit interval which feature a type of power-law nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomialsJn(α,β)(r)withα,β∈(-1,∞),r∈(0,1)andnthe polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity.

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