Teste conv

$f'(x)=\lim _{\Delta x\to 0}{f(x+\Delta x)-f(x) \over \Delta x}$

(Eq. 1)

$S=\int _{a}^{b}f(x)\,dx=\lim _{N\to \infty }\sum _{i=0}^{N}f(x_{i})\Delta x$

$f'(x)=\lim _{\Delta x\to 0}{f(x+\Delta x)-f(x) \over \Delta x^{3}}$

$f'(x)=\lim _{\Delta x\to 0}{f(x+\Delta x)-f(x) \over \Delta x^{2}}$

$x^{3}$

$x^{2}$

$f'(x)=\lim _{\Delta x\to 0}{f(x+\Delta x)-f(x) \over \Delta x}$

$S={\frac {\partial }{\partial t}}\int _{V}f(x,t)=-\int _{S}\mathbf {J} \cdot d\mathbf {s} +\int _{V}S(x,t)dV$

${\frac {\partial }{\partial t}}\int _{V}f(x,t)=-\int _{V}(\nabla \cdot \mathbf {J} )dV+\int _{V}S(x,t)dV$

$\int _{V}\left({\frac {\partial }{\partial t}}f(x,t)+\nabla \cdot \mathbf {J} -S(x,t)\right)dV=0$

${\frac {\partial f}{\partial t}}=-\nabla \cdot \mathbf {J} +S(x,t)$

Lei de Fick:

$\mathbf {J} =-D\;\nabla f$

Onde D é a constante de difusão.

${\frac {\partial f}{\partial t}}=-\nabla \cdot (D\;\nabla f)+S(x,t)$

${\frac {\partial f}{\partial t}}=D\;\nabla ^{2}f+S(x,t)+v{\frac {\partial f}{\partial x}}$

Equação da difusão:

${\frac {\partial f}{\partial t}}=D\;\nabla ^{2}f$

Em uma dimensão:

${\frac {\partial f}{\partial t}}=D{\frac {\partial ^{2}f}{\partial t^{2}}}$

FTCS (Foward Time Central Space):

${\frac {\partial f}{\partial t}}\rightarrow {\frac {f(x,t+\Delta t)-f(x,t)}{\Delta t}}$

${\frac {\partial ^{2}f}{\partial x^{2}}}\rightarrow {\frac {f(x,t+\Delta t)+f(x-\Delta x,t)-2f(x,t)}{\Delta x^{2}}}$

(Escrever a equação em termos numéricos...)

Teste de establilidade do método FTCS:

Um dos modos de Fourier da solução:

$f_{j}^{n}=A^{n}e^{iqjh}$

$h=\Delta x$

$f_{j}^{n+1}=A^{n+1}e^{iqjh}$

$f_{j+1}^{n}=A^{n}e^{iq(j+1)h}$

$f_{j-1}^{n}=A^{n}e^{iq(j-1)h}$

$A^{n+1}e^{iqjh}=k(A^{n}e^{iq(j+1)h}+A^{n}e^{iq(j-1)h}-2A^{n}e^{iqjh})+A^{n}e^{iqjh}$

$k={\frac {D\Delta t}{\Delta x^{2}}}$

$\left|{\frac {A^{n+1}}{A^{n}}}\right|\leq 1$

${\frac {A^{n+1}}{A^{n}}}=1+k(e^{iqh}+e^{-iqh}-2)$

${\frac {A^{n+1}}{A^{n}}}=1+2k[cos(qh)-1]$

${\frac {A^{n+1}}{A^{n}}}=1+4k\;sen^{2}\left({\frac {qh}{2}}\right)$

$\xi =\left|{\frac {A^{n+1}}{A^{n}}}\right|=\left|1+4k\;sen^{2}\left({\frac {qh}{2}}\right)\right|$

Na pior hipótese, o seno quadrado é 1.

$\xi =|1+4k|\leq 1$

$0<k\leq {\frac {1}{2}}$