[Math] How to solve an overdetermined linear system of partial differential equations

finite differenceslinear algebranumerical methodspartial differential equations

I have two partial differential equations that I want to solve (for $\ \sigma $) by finite differences:
$\ -\frac{\partial \sigma}{\partial x}(x,y,t) -p(x,y,t)\frac{\partial \sigma}{\partial t}(x,y,t) = p(x,y,t)$

$\ -\frac{\partial \sigma}{\partial y}(x,y,t) -q(x,y,t)\frac{\partial \sigma}{\partial t}(x,y,t) = q(x,y,t)$

where p and q are known.

This is an over determined system (one unknown, two equations) so I am seeking an Least Squares Error (LSE) solution.

I know that for the inversion problem:

$\ d = \mathbf{F}m $

, where F is a forward operator,
the LSE is:

$\ \mathbf{F}^T\mathbf{F}m = \mathbf{F}^Td$

In my case $\ \mathbf{d}=[p(x,y,t),q(x,y,t)]^T$, $\ \mathbf{m} = [\sigma] $ and
$\ \mathbf{F} =[-\frac{\partial}{\partial x}-p\frac{\partial}{\partial t}, -\frac{\partial}{\partial y}-q\frac{\partial}{\partial t}]^T$ ?

How do I solve this by finite differences?

Thanks in advance for any answers!

EDIT:

Thanks to Andrew we can instead solve the easier equation:

$\ qs_x=ps_y $

for each discrete t value.

I have m discrete x values: $\ x_i, 1<=i<=m $ and n discrete y values: $\ y_j, 1<=j<=n $.

My boundary condition is:

$\ s(m/2,n/2)=t $

I could of course just integrate outwards from the center point, but this would yield a different result depending on the path taken.

Therefore I am looking for a LSE solution to this equation.
How can I do this?

Best Answer

The problem can be simplified a bit. Putting $s(x,y,t)=\sigma(x,y,t)+t$ turns the system into a homogeneous one: $$ -s_x-ps_t=0, $$ $$ -s_y-qs_t=0. $$ Excluding $s_t$ we have $qs_x=ps_y\;$.

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[Math] How to combine two matrix equations into one

I'll show you the code for doing this with square matrices, and leave the generalization to rectangular matrices to you. First we'll generate a data matrix S:

>> n = 4;
>> S=reshape(1:n*n, [n n]);

Now define the shift-by-one matrix on vectors:

>> E = sparse(2:n, 1:n-1, 1, n, n);

and the corresponding central difference matrix:

>> D = (E' - E) / 2;

Now you can define X and Y gradient matrices, which act on matrices (suitably reshaped as vectors) by:

>> I = speye(n);                      % identity matrix
>> Dx = kron(I,D);
>> Dy = kron(D,I);

and use them in the following way:

>> Sx = reshape( Dx * S(:), [n n]);
>> Sy = reshape( Dy * S(:), [n n]);

i.e. whenever you want to calculate a gradient, you first reshape the matrix S into a vector, apply the appropriate derivative operator, and then reshape the result back into a matrix.

Note that I've defined E, D etc as sparse matrices, for computational efficiency. If you want to see what any of these matrices look like, you can convert them to dense matrices using the function full, for example:

>> full(D)
ans =
         0    0.5000         0         0
   -0.5000         0    0.5000         0
         0   -0.5000         0    0.5000
         0         0   -0.5000         0

[Math] finite difference scheme for nonlinear partial differential equations

A little more details as requested (in private email)

Sorry I made a mistake above, of course you want $b \ll 1$. If that is the case you can use whats called "regular perturbation theory" (as opposed to singular perturbation theory which is what you would need if for example the $f_{xx}$ term was multiplied by a small parameter).

Make the following assumption $f(x,t) = f_0(x,t) + b f_1(x,t) + O(b^2)$

Now plug that in and collect terms corresponding to different powers of $b$. You get the following

$$2 \partial_t f_0 - \partial_x f_0 = \partial_x^2 f_0$$

And $f_0$ has the same BCs as you had for $f$ above. This problem is much easier to solve. Then and should give you an approximation when $b \ll 1$. You can even derive a correction by considering the equation you would get for the $f_1$ term. This equation will be forced (i.e. be non-homogenous) by the solution of $f_0$.

Have you looked at any finite difference schemes for parabolic PDEs? Unless you actually try to implement some scheme and run into specific problems, I don't know what you're expceting people on here to tell you about numericas for this thing.

Best Answer

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[Math] How to combine two matrix equations into one

[Math] finite difference scheme for nonlinear partial differential equations

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