## 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?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? 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? pde, pinn, neural network MATLAB Answers — New Questions