Jjjones98

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These are questions asked by Jjjones98

I have the following ODE which I would like to solve with Maple rather than solving by hand (having solved this type of equation by hand many times now):

diff(f(x,y,z),z$2) = A - B*e^(-A*z)

where A and B are constants and I have indicated the second derivative of a function of x,y and z with respect to z.  

I have a differential equation involving several functions of the following form:

diff(h,z) = iAf + iBg,

where h, f and g are functions of the Cartesian coordinates x, y and R and the third coordinate corresponds to z = R for some fixed constant value R.  The derivative is then with respect to the coordinate z and A and B are constants, with i the usual imaginary unit.  Is there some way this equation could be solved explicitly with Maple?

I am working on a physics problem where I will need to generate random unitaries of size N x N.   

As I understand it this would require me to sample uniformly from U(N), where 'uniform' is in the sense of the Haar measure.  I believe this construction in Mathematica is called 'circular unitary ensemble' and was wondering if there was a similar routine in Maple or some simple code that would allow me to generate random unitaries of particular size.

 

I am working on a problem in geometry where I have ended up with a system of nonlinear ODEs in F and G where F and G are functions of a coordinate y, and A and B are both real constants.  I have included the worksheet where I am working on the equations.


Ric_Equations.mw
 

dsolve((2*d^2*F(Y)*F(y)*G(y)/dy^2-(d*F(y)/dy*(d*G(y)/dy))*F(y)+(n-3)*(d*F(y)/dy)^2*G(y))/(4*F(y)*G(y)^2) = A, -(3*(2*d^2*F(y)/dy^2-(d*F(y)/dy)^2*G(y)-(d*F(y)/dy*(d*G(y)/dy))*F(y)))/(F(y)^2*G(y)) = B)

Error, (in dsolve) expecting an ODE or a set or list of ODEs. Received (1/4)*(2*d^2*F(Y)*F(y)*G(y)/dy^2-d^2*F(y)^2*G(y)/dy^2+(n-3)*d^2*F(y)^2*G(y)/dy^2)/(F(y)*G(y)^2) = A

 

NULL

 

 I would like to solve these in as explicit a way as possible by obtaining expressions for F and G and was wondering how best to do this with Maple (it doesn't really matter if the explicit solutions are messy, as I just need to know that they exist and can be written down in some form).  I have tried using the dsolve command directly but I receive the error message which you can see above.

 

 

 

 

 

I am attempting to use the Gram-Schmidt process with Maple to show that the first six orthogonal polynomials which satisfy the following orthogonality condition:

$\int_0^1 (1-x)^{3/2} \phi_n(x) \phi_m(x) dx = h_{n} \delta_{nm}$   

can be expressed in the form:

\phi_0(x) = 1, \phi_1(x) = x − 2/7 , \phi_2(x) = x^2 − (8/11)x + 8/99 , \phi_3(x) = x^3 − (6/5)x^2 + (24/65)x − 16/715 , \phi_4(x) = x^4 − (32/19)x^3 + (288/323)x^2 − (256/1615)x + 128/20995 , \phi_5(x) = x^5 − (50/23)x^4 + (800/483)x^3 − (1600/3059)x^2 + (3200/52003)x − 256/156009.

At the same time I have to find the corresponding values for h_n, so for example, h_0 = 2/5 and h_1 = 8/441.  The polynomials which I obtain have to be combined with the Gaussian quadrature method to show that

$\int_0^1 (1-x)^{3/2} \phi_n(x) \phi_m(x) dx = h_{n} \delta_{nm} \approx \sum_{k=1}^4 c_k f(x_k)$

where x_k are the four roots of \phi_4(x)=0 such that x = [0.0524512, 0.256285, 0.548299, 0.827175]

and the 4 c_k coefficients are given by c = [0.121979, 0.168886, 0.0920439, 0.0170909].

I have learned about Gram-Schmidt orthogonalisation in a basic setting in linear algebra courses where a system of N linearly independent orthogonal vectors is constructed from a system of N linearly independent vectors, but unsure how to apply it to polynomials.  I am also vaguely familar with the idea of appoximating integrals with sets of orthogonal polynomials (Legendre, for example) but not exactly sure how this all works.

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