Abstract
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.
Highlights
Spectral methods are one of the principal methods of discretization for the numerical solution of differential equations
The fundamental goal of this paper is to develop a suitable way to approximate power-law nonlinear ODEs (1) with boundary conditions (2) on the interval (0, 1) numerically using the Jacobi polynomials
We propose a spectral shifted Jacobi-Gauss collocation (SJC) method to find an approximate numerical solution ]N(r)
Summary
Spectral methods (see, for instance, [1,2,3,4,5,6]) are one of the principal methods of discretization for the numerical solution of differential equations. The present problem is rather different from those previously considered in the literature and merits study Another difference is that, in the Lane-Emden problem of the first kind, the power-law index is often an integer. The fundamental goal of this paper is to develop a suitable way to approximate power-law nonlinear ODEs (1) with boundary conditions (2) on the interval (0, 1) numerically using the Jacobi polynomials. For suitable collocation points we use the N − 1 nodes of the shifted Jacobi-Gauss interpolation on (0, 1) These equations together with initial condition generate (N + 1) algebraic equations which can be solved using Newton’s iterative method. We consider a wide variety of parameter regimes, in order to demonstrate the robustness of the numerical scheme Such results are related to radial solutions of the Yamabe problem. We are able to use the present results to infer properties of solutions to the Yamabe problem
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