I was trying to verify my hand solution for this PDE using Mathematica.

Is a trick to help Mathematica obtain solution to $ u_{t}+u=u_{xx}$ with initial conditions $ u(x,0)=f(x)$ and boundary conditions $ u(0,t)=u(L,t)=0$ ?

I tried changing $ L$ to $ 1$ , and tried using explicit $ f(x)$ , but Mathematica still can’t solve it. This is a basic separation of variables PDE which I think Mathematica 11.2 should have been able to solve it. So I thought may be some one can have a trick to help Mathematica with it.

Hand solution

Using separation of variables, let $ u\left( x,t\right) =X\left( x\right) T\left( t\right) $ . Substituting this back into the PDE gives \begin{align*} T^{\prime}X+TX & =X^{\prime\prime}T\ \frac{T^{\prime}}{T}+1 & =\frac{X^{\prime\prime}}{X}=-\lambda \end{align*} Where the separation constant is some real value $ -\lambda$ . This gives the following two ODE’s to solve \begin{align} T^{\prime}+(1+\lambda)T & =0\tag{1}\ X^{\prime\prime}+\lambda X & =0\tag{2} \end{align} Starting with the spatial ODE in order to obtain the eigenvalues. The boundary conditions on the spatial ODE become \begin{align*} X\left( 0\right) & =0\ X\left( 1\right) & =0 \end{align*}

The above boundary value ODE is standard one and its eigenvalues are $ $ \lambda_{n}=\left( \frac{n\pi}{L}\right) ^{2}\qquad n=1,2,3,\cdots $ $ The corresponding eigenfunctions are $ $ X_{n}\left( x\right) =c_{n}\sin\left( \sqrt{\lambda_{n}}x\right) $ $

The solution to the time ODE (1) is, using integrating factor method

$ $ T\left( t\right) =e^{-\left( 1+\lambda_{n}\right) t} $ $

Therefore \begin{align} u\left( x,t\right) & =\sum_{n=1}^{\infty}u_{n}\nonumber\ & =\sum_{n=1}^{\infty}T_{n}\left( t\right) X_{n}\left( x\right) \nonumber\ & =\sum_{n=1}^{\infty}c_{n}e^{-\left( 1+\lambda_{n}\right) t}\sin\left( \sqrt{\lambda_{n}}x\right) \tag{3} \end{align}

In order to determine $ c_{n}$ , the initial condition is now applied. At $ t=0$ , $ u\left( x,0\right) =f\left( x\right) $ and the above becomes $ $ f\left( x\right) =\sum_{n=1}^{\infty}c_{n}\sin\left( \sqrt{\lambda_{n} }x\right) $ $ Multiplying both sides by $ \sin\left( \sqrt{\lambda_{m}}x\right) $ and integrating over the domain of $ f\left( x\right) $ gives $ $ \int_{0}^{L}f\left( x\right) \sin\left( \sqrt{\lambda_{m}}x\right) dx=\int_{0}^{L}\sum_{n=1}^{\infty}c_{n}\sin\left( \sqrt{\lambda_{n}}x\right) \sin\left( \sqrt{\lambda_{m}}x\right) dx $ $ Interchanging the order of summation and integrating gives $ $ \int_{0}^{L}f\left( x\right) \sin\left( \sqrt{\lambda_{m}}x\right) dx=\sum_{n=1}^{\infty}c_{n}\int_{0}^{L}\sin\left( \sqrt{\lambda_{n}}x\right) \sin\left( \sqrt{\lambda_{m}}x\right) dx $ $ By orthogonality of $ sin$ functions, all terms in the right side vanish except when $ n=m$ , leading to \begin{align*} \int_{0}^{L}f\left( x\right) \sin\left( \sqrt{\lambda_{m}}x\right) dx & =c_{m}\int_{0}^{L}\sin^{2}\left( \sqrt{\lambda_{m}}x\right) dx\ & =c_{m}\frac{L}{2} \end{align*} Therefore $ $ c_{m}=\frac{2}{L}\int_{0}^{L}f\left( x\right) \sin\left( \sqrt{\lambda_{m} }x\right) dx $ $ Since $ m$ is an arbitrary number, replacing it back to $ n$ \begin{equation} c_{n}=\frac{2}{L}\int_{0}^{L}f\left( x\right) \sin\left( \sqrt{\lambda_{n} }x\right) dx\qquad n=1,2,3,\cdots\tag{4} \end{equation}

But $ \sqrt{\lambda_{n}}=\frac{n\pi}{L}$ , therefore $ $ c_{n}=\frac{2}{L}\int_{0}^{L}f\left( x\right) \sin\left( \frac{n\pi} {L}x\right) dx\qquad n=1,2,3,\cdots $ $

The above shows that $ c_{n}$ is the Fourier sine series of $ f\left( x\right) $ . Since $ f\left( x\right) $ is not given, explicit solution for $ c_{n}$ can not be found. Therefore the final solution is

\begin{align*} u\left( x,t\right) & =\sum_{n=1}^{\infty}c_{n}e^{-\left( 1+\lambda _{n}\right) t}\sin\left( \sqrt{\lambda_{n}}x\right) \ & =\sum_{n=1}^{\infty}\left( \frac{2}{L}\int_{0}^{L}f\left( x\right) \sin\left( \sqrt{\lambda_{n}}x\right) dx\right) e^{-\left( 1+\lambda _{n}\right) t}\sin\left( \sqrt{\lambda_{n}}x\right) \end{align*}

With $ \lambda_{n}=\left( \frac{n\pi}{L}\right) ^{2}$ .

Mathematica attempts

ClearAll[x, t, u, f]; pde = D[u[x, t], t] + u[x, t] == D[u[x, t], {x, 2}]; ic = u[x, 0] == f[x]; bc = {u[0, t] == 0, u[L, t] == 0}; DSolve[{pde, ic, bc}, u[x, t], x, t] 

ClearAll[x, t, u, f]; pde = D[u[x, t], t] + u[x, t] == D[u[x, t], {x, 2}]; ic = u[x, 0] == x (1 - x); bc = {u[0, t] == 0, u[L, t] == 0}; DSolve[{pde, ic, bc}, u[x, t], x, t] 

ClearAll[x, t, u, f]; pde = D[u[x, t], t] + u[x, t] == D[u[x, t], {x, 2}]; ic = u[x, 0] == x (1 - x); bc = {u[0, t] == 0, u[1, t] == 0}; DSolve[{pde, ic, bc}, u[x, t], x, t] 

Maple 2017.3 is able to solve this

pde:=diff(u(x,t),t)+u(x,t)=diff(u(x,t),x$  2); ic:=u(x,0)=f(x); bc:=u(0,t)=0,u(L,t)=0; pdsolve({pde,ic,bc},u(x,t)) assuming L>0; 

May be next version of Mathematica can solve this?