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
