# Solving a Partio-Integral Differential equation

I had a system of three PDEs $$\frac{\partial \theta_h}{\partial x}+\beta_h (\theta_h-\theta_w) = 0$$

$$\frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c-\theta_w) = 0$$

$$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V\frac{\partial^2 \theta_w}{\partial y^2}-\frac{\partial \theta_h}{\partial x} - V\frac{\partial \theta_c}{\partial y} = 0$$ On eliminating $$\theta_h$$ and $$\theta_c$$ from the third equation I reach $$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} +( -\beta_h - V \beta_c )\theta_w +\beta_h^2 e^{-\beta_h x} \int e^{\beta_h x} \theta_w(x,y) \mathrm{d}x + \beta_c^2 e^{-\beta_c y}\int e^{\beta_c y} \theta_w(x,y)\mathrm{d}y = 0$$ The bc(s) for the system are :

$$\frac{\partial \theta_w(0,y)}{\partial x}=\frac{\partial \theta_w(1,y)}{\partial x}=0$$

$$\frac{\partial \theta_w(x,0)}{\partial y}=\frac{\partial \theta_w(x,1)}{\partial y}=0$$

$$\theta_h(0,y)=1$$$$\theta_c(x,0)=0$$

I must note here that i know an ansatz $$\theta_w(x,y) = e^{-\beta_h x} f(x) e^{-\beta_c y} g(y)$$ which can provide variable separation for the last PDE. But the Eigen value problems that would then come out of this are third order.

Is there any module in MATHEMATICA that can handle Partio-Integral Differential equations ?

Here is something which you can experiment with.

PDE1 = D[Th[x, y], x] + bh*(Th[x, y] - Tw[x, y]) == 0;

PDE2 = D[Tc[x, y], y] + bc*(Tc[x, y] - Tw[x, y]) == 0;

PDE3 = Lh*D[Tw[x, y], {x, 2}] + Lc*V*(D[Tw[x, y], {y, 2}]) -
D[Th[x, y], x] - V*D[Tc[x, y], y] ==
NeumannValue[0, x == 0.] + NeumannValue[0, x == 1] +
NeumannValue[0, y == 0] + NeumannValue[0, y == 1];

bh = 1; bc = 1; Lh = 1; Lc = 1; V = 1; (*Random values*)

sol = NDSolve[{PDE1, PDE2, PDE3, DirichletCondition[Th[x, y] == 1, x == 0],
DirichletCondition[Tc[x, y] == 0, y == 1]}, {Th, Tc, Tw}, {x, 0, 1}, {y, 0, 1}]

Plot3D[Tw[x, y] /. sol, {x, 0, 1}, {y, 0, 1}]

Note: I have presented the solution for the three PDEs before the manipulation which leads to integral PDE.

• Thanks. For some realistic parameters, like bh = 10; bc = 10; Lh = 0.002; Lc = 0.02; V = 1 the code gives me spike behavior at the edges with values of Tw,Th,Tc going into negative . I must mention here that since these are non-dimensional quantities, the minimum value of any solution should be 0 which is happening when parameters you chose are employed. Any way these conditions could be forced here or there is something wrong in my understanding ? – Indrasis Mitra Jan 29 '19 at 5:59
• @IndrasisMitra What conditions? – zhk Jan 29 '19 at 6:04
• conditions such that only accept solutions whose values are not negative .Specifically, since solving the PDE is itself providing negative values for the parameters i wrote, i am at doubt because practically (from the point of view of the system these equations represent) speaking the solution space should not have negative values. – Indrasis Mitra Jan 29 '19 at 6:09
• @IndrasisMitra You need to do parameter sensitivity analysis or recheck your equations and bcs. – zhk Jan 29 '19 at 6:10
• The Dirichlet condition location for Tc in your code needs to be changed from y==1 to y==0 as per the bc(s) from the question. I tried to edit but a minimum 6 character change is needed to qualify as an edit. – Indrasis Mitra Jan 29 '19 at 10:06