\begin{equation} \label{EqBheat.1} \frac{\partial u}{\partial t} = \alpha\, \frac{\partial^2 u}{\partial x^2} \qquad \mbox{or for short} \qquad u_{t} = \alpha\, u_{xx} . \end{equation}

\[ \alpha = \frac{\kappa}{\rho\cdot c_p} , \]

\begin{equation} \label{EqBheat.2} u(x,0) = f(x) , \end{equation}

\[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \qquad\mbox{or for short} \qquad u_t = \alpha^2 u_{xx} , \qquad 0< x < \ell , \quad 0< t < t^{\ast} < \infty , \]

\[ u(0,t) =0 , \qquad u(\ell, t) =0 , \quad t> 0. \]

\[ u(x,0^{+}) = \lim_{t\downarrow 0} u(x,t) = f(x) , \qquad 0< x < \ell . \]

\begin{align*} &\mbox{Heat equation:} \qquad &u_t &= \alpha\,u_{xx} \qquad\mbox{for } 0 < x < \ell \mbox{ and } t > 0, \\ &\mbox{Boundary condition:} \qquad &u (0,t) &= 0 , \quad u (\ell , t) = 0, \qquad t > 0 , \\ &\mbox{Initial condition:} \qquad &u(x,0^{+}) &= \lim_{t\downarrow 0} u(x,t) = f(x) , \qquad 0 < x < \ell , \\ &\mbox{Regularity conditions:} \qquad &f(0) &= u (0,0) = 0 , \quad f(\ell ) = u (\ell , 0) = 0, \qquad \mbox{and} \qquad \lim_{t\to +\infty} u(x,t) = 0. \end{align*}

\[ u_t = \alpha\, u_{xx} \qquad (0< x < \ell) \quad u(0,t) = u(\ell ,t ) =0. \]

\[ u(x,t) = X(x)\, T(t) , \]

\[ X(x)\,\dot{T}(t) = \alpha\, X''(x)\,T(t) \qquad \Longrightarrow \qquad \frac{\dot{T}}{\alpha^2 T(t)} = \frac{X'' (x)}{X(x)} , \]

\[ \begin{split} \dot{T} + \alpha\, \lambda\,T(t) &= 0 , \\ X'' (x) + \lambda\,X(x) &= 0 . \end{split} \]

\[ u(0,t) = X(0)\,T(t) =0 \qquad \Longrightarrow \qquad X(0) =0 , \]

\[ u(\ell ,t) = X(\ell )\,T(t) =0 \qquad \Longrightarrow \qquad X(\ell ) =0 , \]

\[ X'' (x) + \lambda\,X(x) = 0 , \qquad X(0) =0, \quad X(\ell ) =0 . \]

\[ \lambda_n = \left( \frac{n\pi}{\ell} \right)^2 \qquad \Longrightarrow \qquad X_n = \sin \frac{n\pi x}{\ell} , \qquad n=1,2,\ldots . \]

\[ T' (t) + \alpha \left( \frac{n\pi}{\ell} \right)^2 T(t) =0 . \]

\[ u_n (x,t) = X_n (x)\,T_n (t) = e^{- n^2 \pi^2 \alpha\, t / \ell^2} \,\sin \frac{n\pi x}{\ell} , \qquad n=1,2,\ldots . \]

\begin{equation} \label{EqBheat.3} u(x,t) = \sum_{n\ge 1} c_n u_n (x,t) = \sum_{n\ge 1} c_n T_n (t)\, X_n (x) = \sum_{n\ge 1} c_n e^{- n^2 \pi^2 \alpha t / \ell^2} \,\sin \frac{n\pi x}{\ell} . \end{equation}

\begin{equation} \label{EqBheat.4} u(x,0^{+}) = \lim_{t\downarrow 0} \sum_{n\ge 1} c_n u_n (x,t) = \sum_{n\ge 1} c_n u_n (x,0^{+}) = \sum_{n\ge 1} c_n \,\sin \frac{n\pi x}{\ell} = f(x) , \qquad 0< x < \ell . \end{equation}

\[ \sum_{n\ge 1} c_n \,\sin \frac{n\pi x}{\ell} = f(x) , \qquad 0< x < \ell , \]

\begin{equation} \label{EqBheat.5} c_n = \frac{2}{\ell} \int_0^{\ell} f(x)\,\sin \frac{n\pi x}{\ell}\,{\text d}x , \qquad n=1,2,\ldots . \end{equation}

\begin{align*} \frac{\partial u}{\partial t} &= \sum_{n\ge 1} c_n T' (t)\, X_n (x) = \alpha \sum_{n\ge 1} c_n \left( \frac{n\pi x}{\ell} \right)^2 e^{- n^2 \pi^2 \alpha t / \ell^2} \,\sin \frac{n\pi x}{\ell} , \\ \frac{\partial^2 u}{\partial x^2} &= \sum_{n\ge 1} c_n T (t)\, X''_n (x) = - \sum_{n\ge 1} c_n \left( \frac{n\pi x}{\ell} \right)^2 e^{- n^2 \pi^2 \alpha t / \ell^2} \,\sin \frac{n\pi x}{\ell} . \end{align*}

\[ \lim_{x \to 0} \sum_{n\ge 1} c_n T (t)\, X_n (x) = \sum_{n\ge 1} c_n T (t)\, \lim_{x \to 0} \, X_n (x) , \qquad \lim_{x \to \ell} \sum_{n\ge 1} c_n T (t)\, X_n (x) = \sum_{n\ge 1} c_n T (t)\, \lim_{x \to \ell} \, X_n (x) . \]

\[ \lim_{t \to 0} \sum_{n\ge 1} c_n T (t)\, X_n (x) = \sum_{n\ge 1} c_n \lim_{t \to 0} T (t)\, X_n (x) = \sum_{n\ge 1} c_n T (0)\, X_n (x) = \sum_{n\ge 1} c_n X_n (x) = f(x). \]

\[ v_N (x,t) = \sum_{n=1}^N X_n (x) \,T_n (t) = \sum_{n=1}^N u_n (x,t) \]

\[ c_n = \frac{2}{\ell} \int_0^{\ell} f(x)\,\sin \frac{n\pi x}{\ell} \, {\text d} x = - \frac{2\ell}{n^2 \pi^2} \int_0^{\ell} f''(x)\,\sin \frac{n\pi x}{\ell} \, {\text d} x . \]

\[ \frac{\mbox{Rate}\ \mbox{ of }\ \mbox{ heat}\ \mbox{ transfer}}{\mbox{area}} = -\kappa\,\nabla u = - \kappa\,\frac{\partial u}{\partial x} , \]

\[ \left. \frac{\partial u}{\partial x} \right\vert_{x=0} = 0 \qquad \mbox{and} \qquad \left. \frac{\partial u}{\partial x} \right\vert_{x=\ell} = 0, \] reflecting the fact that there will be no heat transferring, spatially, across the points x = 0 and x = ℓ. The heat conduction problem becomes the initial-boundary value problem below. The first step is to temporarily disregard the initial condition (because it is non-homogeneous) and consider an auxiliary boundary value problems. Next, we apply the separation of variables for the function u(x, t) = X(x) T(t )that leads to two ordinary differential equations where primes stand for derivatives (Lagrange's notation). The homogeneous Neumann boundary conditions are also separated into This leads to the eigenvalue problem Let us first show that λ = 0 is an eigenvalue. Indeed, the general solution of the differential equation with this λ is To satisfy the Neuman boundary conditions, we have to set c₂ = 0. Then function \( X(x) = c_1 \) is a solution of the differential equation \( X'' (x) + 0\cdots X(x) = 0 \) and automatically satisfies both Neuman boundary conditions. Therefore, any constant solution X(x) = c₁ ≠ 0 is an eigenfunction.

Assuming λ to be positive, the general solution of the differential equation for X(x) becomes

with some constants c₁ and c₂. Taking derivative \( X'(x ) = \sqrt{\lambda} \left[ -c_1 \sin \left( \sqrt{\lambda}\,x \right) + c_2 \cos \left( \sqrt{\lambda}\,x \right) \right] , \) and substituting into the boundary conditions, we get Since λ is positive (by assumption), the only choice we have is to set \( \sin \left( \sqrt{\lambda}\,\ell \right) =0 \) because we cannot affort a trivial solution by setting c₂ = 0. Then we obtain the eigenvalues and corresponding eigenfunctions: We included n = 0 because λ = 0 is an eigenvalue with a constant as an eigenfunction. Every auxiliary function u_n(x, t) = X_n(x) is a solution of the homogeneous heat equation \eqref{EqBheat.1} and satisfy the homogeneous Neumann boundary conditions. Thereofre, any their linear combination will also a solution of the heat equation subject to the Neumann boundary conditions. Hence we expect the sum over all auxiliary partial nontrivial solutions will be a solution of our given IBVP provided that the infinite series converges. So we seek the solution in the form: where coefficients c_n are obtained from expansion of f(x) into Fourier cosine series: Example 2A: heat conduction problem with Neumann boundary conditions Example 2B: heat conduction problem with Neumann boundary conditions and discontinuous initial condition Example 2C: heat conduction problem with eigenfunction as initial distribution

 

We consider two initial boundary value problems with Dirichlet's condition (when temperature is prescribed to be zero) on one end and Neumann condition (which is equivalent insulation) on another end.

Problem DN:

When regularity conditions are satisfied, the given initial boundary value problem (IBVP for short) is well-posed. However, when the conditions at end points do not meet or the initial temperature is modelled by a ppiecewise continuous function, the IBVP can also be solved, but it becomes ill-posed and numerical calculations must be done with care.

Disregarding the initial condition, we obtain an auxiliary boundary value problem. We seek for its non-trivial solutions that are represented as the product u(x, t) = X(x) T(t) of two functions, X(x) and T(t), each of which depends on single independent variable. Substituting this form into the heat equation, we get

Its left-hand side depends on time variable t, but the right-hand side depends on spatial variable x that are independent. Therefore, this equation holds only when both sides are constants. As usual, we denot this constant as −λ. This give two differential equations containing a parameter λ (which is unknown yet): From the homogeneous boundary conditions, we get and The boundary conditions for X(x) together with the second order differential equation form so called the Sturm--Liuville problem: It can be shown that the Sturm--Liouville problem has nontrivial solutions only for positive values of parameter λ. Assuming this, we obtain the general solution to the differential equation \( X'' (x) + \lambda\,X(x) =0 \) to become To satisfy the boundary condition at x = 0 we have to set c₁ = 0 and obtain \( X(x) = c_2 \sin \left( \sqrt{\lambda}\,x \right) . \) An arbitrary constant c₂ ≠ 0 in order to avoid the identically zero solution, so it can be dropped. Next substitution of this function into the boundary condition at x = ℓ, we get because λ > 0 (by assumption) and c₂ ≠ 0 to avoid trivial solution. Therefore, λ must be a root of the transcendent equation With these eigenvalues, we obtain eigenfunctions to be Substituting λ = λ_N into the differential equation for T(t), \( T' (t) + \alpha\lambda\, T(t) = 0 , \) we get with some arbitrary constant c_n. Multiplying T_n(t) and X_n(x), we find partial nontrivial solutions to our auxiliary boundary value problem. Upon summing all these partial nontrivial solutions, we represent the required solution as an infinite series Every term in this infinite series satisfies the heat conduction equation and both boundary conditions. Assuming that this series converges, we extend this statement for the series. Now substituting u(x, t) into the initial condition, Equating the right-hand side to f(f), we obtain This identity is valid only when the limit can be evaluated under the sum sign (which is true when the series converges uniformly). It is not our case, and we apply the Cesàro summation to prove the validity of interchanging the order of limiting procedure and summation. The coefficients in this expansion are known to be Example 3A: heat conduction problem with mixed boundary conditions Example 3B: heat conduction problem with mixed boundary conditions and constant initial temperature

Problem ND: We consider a heat transfer problem for a rod of length ℓ with Dirichlet's condition (when temperature is prescribed to be zero) on right end x = ℓ and Neumann's condition (which is equivalent insulation) on another end x = 0.

Application of separation of variables leads to the Sturm--Liouville problem It has eigenfunctions So the heat equation subject to the homogeneous boundary conditions has a sequence of nontrivial solutions: Now solution of the given IBVP is the sum of all partial nontrivial solutions where the coefficients are Example 3C: heat conduction problem with mixed boundary conditions Example 3D: heat conduction problem with mixed boundary conditions and piecewise continuous initial temperature

 

Problem 4A: We start with the initial boundary value problem:

So the left end x = 0 is maintained with a constant (zero) temperature, while on the another end x = ℓ there is a convection heat transfer with surrounded air of temperature T _∞. Since we are in the section to illustrate separation of variables method, we make a simplified assumption that T _∞ is zero. This leads to the following (simplified boundary conditions: where γ is a positive constant. In order to apply the separation of variables method, we temporarily disregard the initial condition (because it it is nonhomogeneous) and obtain the auxiliary boundary value problem Assuming that this problem has a solution represented as a product of two functions, u(x, t) = X(x) T(t), we obtain upon its substitution into the heat equation that Substituting the product u(x, t) = X(x) T(t) into the boundary conditions, we obtain and So together with the differential equation, we obtain the Sturm--Liouville problem Assuming that λ is posityive (for other values of λ, the corresponding Sturm--Liouville problem has no nontrivial solution), we obatian solution that satisfies the boundary condition at x = 0: Substitution into the boundary condition at x = ℓ yields Introducing a new parameter μ = λ^½ℓ, we reduce the equation \eqref{Eqconvert.1} to the following transcendent equation Let μ_n (n = 1, 2, …) be a sequence of positive zeroes of Eq.\eqref{Eqconvert.2} (μ = 0 is not an eigenvalue because the corresponding eigenfunction is identically zero). Then eigenvalues and eigenfunctions of the Sturm--Liouville problem are We seek its solution in the form: where T_n(t) is and X_n(x) are eigenfunctions for the Sturm--Liouville (SL) problem The solution of the given IBVP is represented as an infinite series Example 4A: heat convention problem

Problem 4B:

Example 4B: heat convention problem

Problem 4C:

Example 4C: heat convention problem Example 5: heat conduction problem with convection boundary conditions \( (-\infty , \infty ) . \)

 

The Heat Equation: Inhomogeneous Boundary Conditions
https://medium.com/cantors-paradise/the-heat-equation-a76d7773a0b5
spike.gif FourierNumber.gif
https://medium.com/cantors-paradise/the-heat-equation-forcing-65af3a82c8b2
https://medium.com/cantors-paradise/the-heat-equation-nonhomogeneous-boundary-conditions-eea5910d777c
https://faculty.math.illinois.edu/~laugesen/286/heat_eq.pdf
http://www.math.ttu.edu/~gilliam/ttu/s10/m3351_s10/c13_Sep_Vars_Heat_Ex.pdf
https://mathematica.stackexchange.com/questions/138954/how-to-solve-1d-heat-equation-with-neumann-boundary-conditions
https://www.mn.uio.no/ifi/english/services/knowledge/cs/forelesn.kap7.pdf
https://www.sciencedirect.com/topics/engineering/neumann-boundary-condition
https://tutorial.math.lamar.edu/classes/de/solvingheatequation.aspx
derivation:
https://projecteuclid.org/download/pdf_1/euclid.ade/1355867704
https://web.stanford.edu/class/math220b/handouts/heateqn.pdf

Python:
http://folk.ntnu.no/leifh/teaching/tkt4140/._main056.html
http://hplgit.github.io/num-methods-for-PDEs/doc/pub/diffu/sphinx/._main_diffu001.html
https://fenicsproject.org/pub/tutorial/html/._ftut1006.html

Maple (with animation):
https://www.maplesoft.com/applications/view.aspx?SID=3601&view=html Example: A temperature “spike” Example: heat conduction problem with Dirichlet's conditions Example: heat conduction problem with harmonic initial conditions Example: heat conduction problem with Neumann's boundary conditions Example: heat conduction problem with mixed boundary conditions

 

References

 

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