How to formulate this eq. (system) with matrices and vector-valued functions?

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Question:How to formulate this eq. (system) with matrices and vector-valued functions?

Posted: Pinetree 15

December 04 2017

Apologies for possible double post, it seemingly locked up upon trying to post the first time.

So before I get as far as to ask for how to get a certain PDE system solved and plotted, I tried to fiddle a little bit around with linear algebra.

First, the file SystemGoesWrong.mw . I have issues with declaring (same result if I remove the with(VectorCalculus)); as far as I can see, EQ0 and EQ00 should be the same, except that I have summed the vector in one of them. And the first that "works", is wrong: it returns a scaling of a vector. How come?
But then I copy everything from the heading and down into a worksheet where I was already fighting some linear algebra things (can someone please explain?): SystemDeclaresBut.mw
Then EQ0 and EQ00 declare just fine! What is the issue?
How do I get Maple to list the equations in "compact" form with vector-valued functions so that I can read and debug? The actual PDE system I want to solve (numerically, of course), looks as follows: DE4Maple.pdf
That was also the reason why I tried to declare procedures (coordinate-wise maximum ...), but I guess that questions on how to extract a solution and plot it in a particular way will be its own posting after I have learned how to declare it.

The contents of the first file:

# Since I do not have any idea of how to get vectors nicely, ...

... I replace U0, U1, U2 by u,v,w and use difftables U,V,W. And y1, y2 replaced by y,z.

(1)

M := Matrix(2, 2, {(1, 1) = 2, (1, 2) = -1, (2, 1) = -1, (2, 2) = 2})

(2)

Matrix([[2/3, 1/3], [1/3, 2/3]])

(3)

# The system

(4)

EQ00 := `<|>`(F1, F2).(1/M.`<,>`(q1, q2)-`<,>`(U[y], V[z]))-(1/4)*LinearAlgebra:-Transpose(1/M.`<,>`(q1, q2)-`<,>`(U[y], V[z])).(1/M.`<,>`(q1, q2)-`<,>`(U[y], V[z]))-1/2*(2*`<|>`(F1, F2)+`<|>`(U[y], V[z])-`<|>`(s1, s2)).(1/M).`<,>`(q1, q2)

Error, (in rtable/Sum) invalid input: dimensions do not match: Matrix(1 .. 1, 1 .. 2) cannot be added to Vector[row](1 .. 2)

Download SystemGoesWrong.mw

... and of the second:

Over to some linear algebra.

(1)

(2)

Matrix([[6, 8], [9, 12]])

(3)

(4)

normsqbyDot := proc()
local theArgs, arg, look, me, cf, term;
   1   me := eval(procname,1);
   2   theArgs := args;
   3   look := tablelook(('procname')(theArgs),'[`/POS`(1,normsqbyDot,1), `/BIND`(1,1,`/y1`::VectorCalculus:-Vector), `/PATTERN`(`/y1`^2)]');
   4   if look <> FAIL then
   5     eval(look,`/FUNCNAME` = procname)
       else
   6     ('procname')(theArgs)
       end if
end proc

normsqbyMatrixProduct := proc()
local theArgs, arg, look, me, cf, term;
   1   me := eval(procname,1);
   2   theArgs := args;
   3   look := tablelook(('procname')(theArgs),'[`/POS`(1,normsqbyMatrixProduct,1), `/BIND`(1,1,`/y1`::Matrix), `/PATTERN`(`/y1`^2)]');
   4   if look <> FAIL then
   5     eval(look,`/FUNCNAME` = procname)
       else
   6     ('procname')(theArgs)
       end if
end proc