There is no theorem which tells us about polynomial approximation of a BVP. However, if you choose to apply Collocation method, select polynomials which are zeros on the boundary and select roots of a  orthogonal polynomial as collocation points.

There is no theorem which tells us about polynomial approximation of a BVP. However, if you choose to apply Collocation method, select polynomials which are zeros on the boundary and select roots of a  orthogonal polynomial as collocation points.

Generally speaking, expansion in terms of the Chebyshev polynomials of the first kind is the best. For this, naturally the interval of interest should first be transformed into the interval [-1,1] (this is achieved by means of a linear transformation -- a scaling followed by an affine transformation) where the latter polynomials are defined (I am assuming that you are dealing with an ordinary non-linear differential equation over a one-dimensional domain). Part of the attraction of the Chebyshev polynomials lies in the fact that evaluation of the recursive relations of polynomials that vary between [-1,1] over the interval [-1,1] is numerically (using floating-point operations) very stable so that one can use very high-order expansions without suffering from numerical instability.

For completeness, I attach a relevant text.

Incidentally, in dealing with differential equations, whether linear or non-linear, one can also use a variety of spline approximations (for example cubic splines), which fit the solution locally; the full solution consists of an appropriate matching of the splines at the boundaries of the sub-intervals over which the splines are defined (the higher the order of the splines, the more derivatives are matched at the boundaries of the relevant sub-intervals).

http://www.mathematik.uni-kassel.de/koepf/cheby.pdf

(i)Using Chebyshev polynomials in MATLAB ,you will get some idea to solve your problem.

(ii)  use null work/ virtual work approach  approach in PRL and Proc.Ind.Nat.Sci.Aca-A

Good luck 

Hello,

in case of BVP, you should look at the continuous solution or dense output of ODE solver such as in the case of Runge-Kutta methods. In particular, the work of Lawrence F. Shampine about the solver bvp4c (cf http://faculty.smu.edu/shampine/finalbvp5c.pdf).

Cheers

Many applications apply Chebyshev polynomials. However, you can take into account some specific characteristic of your problem to explore the polynomial representation. See the paper: https://www.ipen.br/biblioteca/1999/07366.pdf 

It strongly depend on the character of nonlinearity. We have to remember than nonlinear PDE are solved by iterative proces. So we have to use as simple polynomial as possible. The MATLAB is a good choice.

It depends on the numerical method you are using for the approximations. In finite element you need to study the ellipticity of your bilinear forms. My advise is choose approximations polynomial to be conforming with the governing BVP. If the unknown variables are of C^2 choose the approximation space to be in H^2 or C^2.

I recommend that It is best to use the shifted Legendre and shifted second kind Chebyshev polynomials which are orthogonal and orthonormal and with compact support.

R.RAJARAMAN

DEPARTMENT OF MATHEMATICS

SASTRA UNIVERSITY

THANJAVUR

INDIA