Approximation of the heat equation: Fourier method - Historial de revisiones

Carlos Castro: /* MATLAB code */

2016-04-26T09:26:17Z

‎MATLAB code

2016-04-26T09:13:25Z

‎MATLAB code

2016-04-26T09:05:25Z

‎Example:

2016-04-26T09:04:26Z

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2016-04-26T09:03:21Z

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2016-04-26T08:58:47Z

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2016-04-26T08:57:59Z

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2016-04-26T08:56:51Z

2016-04-26T08:49:03Z

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2016-04-26T08:48:26Z

@@ Línea 248: / Línea 248: @@
 u=c(1)*exp(-pi^2/4*tt).*sin(pi*xx/2);
 for k=2:M
-    u=u+c(k)*exp(-k^2*pi^2/4*tt).*sin(k*pi*xx);
+    u=u+c(k)*exp(-k^2*pi^2/4*tt).*sin(k*pi*xx/2);
 end
 figure(2)

@@ Línea 217: / Línea 217: @@
 % Fourier series approximatin
 clear all
 % Number of Fourier coefficients M
 M=5;
-% Define the function that we are going to approximate
+% space discretization
-f=(1-2*abs(x-1/2));
+N=100; dx=(b-a)/N;
-% Write the trigonometric basis in pn
+% Compute the Fourier coefficients
-% and compute the Fourier coefficients an
-p=zeros(M,length(x));
 aproxima=0;  % aproximate function
 for k=1:M
-    p(k,:)=sin(k*pi*x);
+    p=sin(k*pi*x/2); % eigenfunction
-    a(k)=trapz(x,f.*p(k,:))/trapz(x,p(k,:).^2);
+    c(k)=trapz(x,u0(x).*p)/trapz(x,p.^2);
-    aproxima=aproxima+a(k)*p(k,:);
+    aproxima=aproxima+c(k)*p;
 end
 % Draw the function and the approximation
 figure(1)
-plot(x,[aproxima;f])
+plot(x,[aproxima;u0(x)])
 % Draw solutions of heat equation
 dt=1/100; % time mesh step
 t=0:dt:2; % time mesh
 [xx,tt]=meshgrid(x,t);
-u=a(1)*exp(-pi^2*tt).*sin(pi*xx);
+u=c(1)*exp(-pi^2/4*tt).*sin(pi*xx/2);
 for k=2:M
-    u=u+a(k)*exp(-k^2*pi^2*tt).*sin(k*pi*xx);
+    u=u+c(k)*exp(-k^2*pi^2/4*tt).*sin(k*pi*xx);
 end
 figure(2)

@@ Línea 207: / Línea 207: @@
 </math>
-that we substitute into:
+and therefore
 <math>
-w_Q(x,t)=\sum_{k=1}^{Q} {T_k(t)}\sin \frac{k \pi x}{2},
+w_Q(x,t)=\sum_{k=1}^{Q} c_k e^{-\frac{k^2 \pi^2}{4}  t}\sin \frac{k \pi x}{2},
 </math>

@@ Línea 202: / Línea 202: @@
 The solution of this equation is:
 <math>
-T_k(t)&=&c_k e^{-\frac{k^2 \pi^2}{4}  t}, \hspace{0.4cm} t \in [0,3],
+T_k(t)=c_k e^{-\frac{k^2 \pi^2}{4}  t}, \hspace{0.4cm} t \in [0,3],
 </math>
 that we substitute into:
 <math>
 w_Q(x,t)=\sum_{k=1}^{Q} {T_k(t)}\sin \frac{k \pi x}{2},

@@ Línea 189: / Línea 189: @@
 <math>
-w_Q(x,t)=\sum_{k=1}^{{Q}} {T_k(t)}\sin \frac{k \pi x}{2},
+w_Q(x,t)=\sum_{k=1}^{Q} {T_k(t)}\sin \frac{k \pi x}{2},
 </math>
 then substituting in the heat equation we obtain an equation for each Fourier coefficient.
 [[Categoría:Grado en Ingeniería Civil y Territorial]]

@@ Línea 152: / Línea 152: @@
 <math>
 \left\{ \begin{array}{ll}
-y''+ \lambda y =0, \hspace{0.4cm} x \in (0,2), \\ y(0)=y(2)=0.
+\varphi''+ \lambda \varphi =0, \hspace{0.4cm} x \in (0,2), \\ \varphi(0)=\varphi(2)=0.
 \end{array}   \right.
 </math>

@@ Línea 53: / Línea 53: @@
 u_k (x,t)= e^{-k^2\pi^2 /L^2 t}\sin(\frac{k\pi x}{L})
 </math>
 Thus, we only have to deal with the initial conditions. We consider two different cases:
-'''1. The initial condition is a linear combination of eigenfunctions'''
+==The initial condition is a linear combination of eigenfunctions==
 We assume that we can write the initial condition <math>u_0(x)</math> as
@@ Línea 74: / Línea 72: @@
 is a solution of the full system: heat equation, boundary conditions and initial conditions.
-===Example ===
+'''Example'''
 Consider the problem
@@ Línea 113: / Línea 111: @@
 }}
-'''2. The initial condition is not a finite linear combination of eigenfunctions'''
+==The initial condition is not a finite linear combination of eigenfunctions==
 In this case, the main idea is to approximate the initial condition <math>u_0(x)</math> by
@@ Línea 136: / Línea 134: @@
 </gallery>
-===Example: ===
+==Example: ==
 Consider the problem
@@ Línea 147: / Línea 145: @@
 We solve it by the Fourier method
 [[Categoría:Grado en Ingeniería Civil y Territorial]]