How to correct set conditions and params of PDE?

Dear members of community! I have a important problem with PDE Toolbox initialization coeffs and conditions.

I try to solve heat equation and compare exact solution and PINN solution looks like Solve Poisson Equation on Unit Disk Using Physics-Informed Neural Networks - MATLAB & Simulink (mathworks.com) . But, first I should correct create mesh and for obtain solution (numerical solution).

Problem formulation:

Consider the next mathematical problem: heat equation with initial and boundary conditions - modes with exacerbation. Let we have specific auomdel heat equation

satisfies boundary an initial conditions:

Automodel general solution of this problem is:

where is solution of ODE problem:

The solutions of this problem for is

Matlab code:

Create the PDE model and include the geometry.

model = createpde;
R1 = [3,4,0,1,1,0,0,0,10,10]';
g = decsg(R1);
geometryFromEdges(model,g);
pdegplot(model,EdgeLabels="on")
axis equal
grid on

Define constants of PDE and initial and boundary equations:

k0 = 1;  % Adjust as necessary
sigma = 2;  % Adjust as necessary
A0 = 2;  % Adjust as necessary
T = 0.5;  % Adjust as necessary
n = 2;  % Adjust as necessary

% Initial conditions
setInitialConditions(model,0);

% Boundary conditions
applyBoundaryCondition(model, 'dirichlet', 'Edge', 1, 'u', @(region,state) A0 * (T - state.time)^n);

% PDE coefficients
specifyCoefficients(model, 'm', 0, 'd', 1, 'c', @(region, state) k0 * state.u.^sigma, 'a', 0, 'f', 0);

% Generate mesh
generateMesh(model, 'Hmax', 0.1);

Try to obatin numerical solution:

% Solve the PDE
tlist = linspace(0, T, 50);
result = solvepde(model, tlist);
u = result.NodalSolution;

I understand, that obtainded numerical solution is not correct, and training PINN using this meshes and PDE coeffs non coorrect step os obtain solution:

Is not correct solution.

My problem:

How to correct set intial and boundary conditions, and create geometric dash for solve this PDE?

[Torsten]

You can't solve PDEs over infinite domains in x and t with the PDE toolbox.

[Alex Milns]

If consider infinity, for example, 10e3, can I use this toolbox? I understand that an infinite region cannot be specified in this case

[Torsten]

In principle, you only have a boundary condition at x = 0. Further, the boundary conditions at t=-oo and the boundary condition at x = 0 are not compatible for n > 0 since A0*(t-T)^n should converge to 0 as t->-Inf.

The PDE toolbox expects a closed bounded region where you can define a boundary condition on each of its edges. Can you fulfill this requirement for your problem ? I don't see how.

[Alex Milns]

1) I'm sorry, I just realized that n should be negative (for example n = -1.25)

2) The only thing in this case to which I can reduce this problem is to make the second boundary condition similar and consider a certain region, not at infinity, but with a fractional dimension of sigma -- then this will also be a self-similar regime, although in the analytical solution then in In this case, a minus sign will appear in front of the 'x'

[Torsten]

1) So you have a pole at t = T ?

2) I don't understand it.

Why do you want to force the PDE Toolbox to solve a problem you already know the solution of ?

[Alex Milns]

I want to compare PINN solution and numerical, it is posibble, if I correctly define mesh (domain) and obtain numnerical solution, close to exact...

This problem was not formulated by me, and in this automodel behaviour there is no condition at t = T!

Perhaps, PINN not be able to solve this problem and PDE toolbox not work in this case (