How can I use Createpde to solve a fourth degree partial differential equation?

Hello,

How can I use Createpde to solve the a fourth degree partial differential equation?

∇4(u)=q/D?

[Torsten]

∇2(u)=v

∇2(v)=q/D?

[Poupak Kermani]

Thank you for your feedback.

I have created the first part of the solution (i.e.: ∇2(v)=q/D ):

% Rectangle is code 3, 4 sides,
% followed by x-coordinates and then y-coordinates
R1 = [3,4,-1,1,1,-1,-1,-1,1,1]';
% Circle is code 1, center (.5,0), radius .2
C1 = [1,0.,0,.2]';
% Pad C1 with zeros to enable concatenation with R1
C1 = [C1;zeros(length(R1)-length(C1),1)];
geom = [R1,C1];
% Names for the two geometric objects
ns = (char('R1','C1'))';
% Set formula
sf = 'R1 - C1';
% Create geometry
g = decsg(geom,sf,ns);
model = createpde(2);
geometryFromEdges(model,g);
pdegplot(model,EdgeLabels="on")
xlim([-1.1 1.1])
axis equal;

%
applyBoundaryCondition(model,"dirichlet", ...
    Edge=3,u=[1,1]);
applyBoundaryCondition(model,"dirichlet", ...
    Edge=1,u=[1,1]);
applyBoundaryCondition(model,"dirichlet", ...
    Edge=4,u=[1,1]);
applyBoundaryCondition(model,"dirichlet", ...
    Edge=2,u=[1,1]);
% The next step is optional,
% because it sets "g" to its default value
applyBoundaryCondition(model,"neumann", ...
    Edge=5:8,g=[0,0]);
%
%
figure;
specifyCoefficients(model,m=0,d=0,c=1,a=0,f=[1;1]);
generateMesh(model,Hmax=0.1);
results = solvepde(model);
v = results.NodalSolution;
pdeplot(model,"XYData",v)

●

Could you guide me on how I could create the second part of the solution (i.e.: ∇2(u)=v )?

Answer 1

[Torsten]

You have a system of two second-order differential equations. Thus instead of defining one equation, you have to define two and solve them simultaneously.

I don't have experience with the PDE toolbox - thus I can only tell you what you should do, not how you can do it.

But I think this is almost what you need:

https://uk.mathworks.com/help/pde/ug/clamped-square-isotropic-plate-with-a-uniform-pressure-load.html?#responsive_offcanvas

[Poupak Kermani]

Hello,

Thank you very much for your feedback.

You have helped me greatly to solve my problem. Your last comment unblocked this issue.

Best regards,

Poupak