# Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations

$$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ \partial x^2} +f(x)\\ \frac{\partial T_2(x,t)}{\partial t}&=& \alpha_2 \frac{\partial^2 T_2(x,t)}{ \partial x^2} \end{eqnarray}$$ with f(x) an exponential source term, the initial and boundary conditions:

$$\begin{eqnarray} T_1(0,0)&=&T_0,(x->0)\\ T_2(l,0)&=&T_0,(x->l)\\ T_1(l,s)&=&T_2(l,s),(x->l)\\ T_1'(l,s)&=&K_\lambda T_1'(l,s),(x->l)\\ T_2(l,0)&=&T_0,(x->l)\\ T_2(\infty,s)&=&\frac{T_0}{s},(x->\infty)\end{eqnarray}$$

Please do not hesitate with any kind of suggestions. After a Laplace transform, to both equations, I stuck to find constant C1, C2, C3. by the mean of this short code (evaluation doent give any result, I do not know why !! $$\begin{eqnarray} T_{1}[x,s]:=C_{1}exp{-\lambda x}+ c_{2}exp{-\lambda x}+\frac{A}{s}+\frac{T_{0}}{s}\\ T_{2}[x,s]:=C_{3}exp{-\beta x}+\frac{T_{0}}{s}\\\end{eqnarray}$$ $$\begin{eqnarray} BC1=k_{1}T'_{1}[x,s]==0/. x->0;\\ BC2=T_{1}[l,s]==T_{2}[l,s];\\ BC3=(k_{1}T'_1[x,s]-k_{2}T'_{2}[x,s]==0/. x->l;\\ BC4=k_{2}T'_{2}[x,s]==0 /. x->L; \end{eqnarray}$$

$$\begin{eqnarray}eqns&=Flatten[{BC1,BC2,BC3,BC4}];\\ var&={C_{1},C_{2},C_{3}};\\ soln&=Simplify[Solve[eqns,var]]\\\end{eqnarray}$$

Where the origin is at x=0, the interface at x=l and the end bar at x=L. Have you another method ? Merci beaucoup.

## migrated from mathematica.stackexchange.comAug 30 '15 at 11:39

This question came from our site for users of Wolfram Mathematica.

• As written the equations are not coupled, but one matching boundary condition is. Solving the equations and applying the conditions should be straight forward by most processes. The issue is the matching b.c. which provides a connection between the coefficients of the solutions. Also, what is $K_{\lambda}$ ? – Leucippus Aug 30 '15 at 14:19
• $K_{\lambda}=\frac{k_1}{k_2}$ is the ratio of thermal conductivity. What you said is right, my problem now is to find constants satisfying b.c. – user265521 Sep 1 '15 at 21:44