60 lines
1.5 KiB
Mathematica
60 lines
1.5 KiB
Mathematica
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function [x,t,u] = Dirichlet_EA(L,N,T,K,c1,c2,f,u0,D)
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% ---- Solution of heat equation ----
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% u_t - u_xx = f in the interval [-L,L] (doesn't matter if we change it)
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% Dirichlet BC
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%
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% -----------------------------------------------
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% Sintassi:
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% [x,t,u]=calore_template(L,N,T,K,c1,c2,fun,u0)
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%
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% Input:
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% L Half of the width (-L,L)
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% N number of intervals in (-L,L)
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% T max time (0,T)
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% K number of intervals in (0,T)
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% c1 Dirichlet BC in x=-L
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% c2 Dirichlet BC in x=L
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% f force function
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% u0 initial condition in t=0
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%
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% Output:
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% x vector of the spatial nodes
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% t vector of the time nodes
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% u numeric solution of the problem
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% discretisation step in time and space
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h=L/N; %space
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tau=T/K; %time
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% Initialization of t
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t=linspace(0,T,K+1)';
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% initialization of x
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x=linspace(0,L,N+1);
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% Initialization of the matrix solution u
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u=zeros(N+1,K+1);
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% Intial conditions
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u(:,1)=u0(x);
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% BC
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u(1,:)=c1(t);
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u(end,:)=c2(t);
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% Creation of the matrix A
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% We obtained this matrix using the Central Discretization method
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e=ones(N-1,1);
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A=spdiags([-e,2*e,-e],[-1,0,1],N-1,N-1)/(h^2);
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I=speye(N-1,N-1);
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% Compute the solution
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% What we are doing here is to compute the solution in the interval for
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% each time k
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for k=1:K
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% Assembly of the force
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F=f(x(2:end-1),t(k));
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% Correction of the force using the BC
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F(1)=F(1) + D*c1(t(k))/(h^2);
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F(end)=F(end) + D*c2(t(k))/(h^2);
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% Solution using Eulero forward
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u(2:end-1,k+1) = ((I - tau*D*A)*u(2:end-1,k))' + tau*F;
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end
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