General analytical solution of the boundary value problem

In this chapter we will construct the general solution of the heat conduction boundary value problem by the method of separation of variables (the Fourier method). But first, a few introductory remarks. We want to solve the inhomogeneous equation (1.8$\displaystyle \frac{\partial T(M, t)}{\partial t}$ $\displaystyle = a^2 \cdot \Delta T(M, t)$ $\displaystyle + f(M, t),$) with the initial condition $T_0(M)$ and inhomogeneous boundary conditions (1.13$\displaystyle T(M, t)$ $\displaystyle = \Phi(M, t), \quad M$ $\displaystyle \in S.$) and (1.14$\displaystyle \lambda \cdot \frac{\partial T(M, t)}{\partial \vec{n}}$ $\displaystyle = \Phi(M, t), \quad M$ $\displaystyle \in S,$). Solving it head-on can be very difficult, so first we have to get rid of the inhomogeneity of the boundary conditions.

Let us represent the solution as the sum $T(M, t)$ $= \widehat{T}(M, t)$ $+ U(M, t)$, where $\widehat{T}(M, t)$ is the solution of the problem with homogeneous boundary conditions, and $U(M, t)$ is a function that will absorb the inhomogeneity of the boundary conditions. Let us substitute the sum into equation (1.8$\displaystyle \frac{\partial T(M, t)}{\partial t}$ $\displaystyle = a^2 \cdot \Delta T(M, t)$ $\displaystyle + f(M, t),$) and obtain

The form of the function $U(M, t)$ we will determine later, when considering specific problems; for now let us note that it can be chosen as a polynomial defined up to coefficients, which are found from the boundary conditions. The new heat source density function has the form

The coefficients $U(M, t)$ for the Dirichlet boundary conditions are found by substituting the solution into the boundary conditions (1.13$\displaystyle T(M, t)$ $\displaystyle = \Phi(M, t), \quad M$ $\displaystyle \in S.$). The solution $\widehat{T}(M, t)$ satisfies the homogeneous Dirichlet condition, that is, vanishes on the boundary $S$, whence

Similarly, the coefficients $U(M, t)$ for the Neumann boundary conditions are found by substituting the solution into the boundary conditions (1.14$\displaystyle \lambda \cdot \frac{\partial T(M, t)}{\partial \vec{n}}$ $\displaystyle = \Phi(M, t), \quad M$ $\displaystyle \in S,$). The solution $\widehat{T}(M, t)$ satisfies the homogeneous Neumann condition, that is, its normal derivative on the boundary $S$ vanishes, whence

Let us not forget the initial condition, which does not compute any coefficients but is simply rewritten as

After computing the coefficients of the polynomial function $U(M, t)$ the inhomogeneous boundary value problem with inhomogeneous boundary conditions reduces to an inhomogeneous boundary value problem with homogeneous boundary conditions. Further, to simplify the notation somewhat, we will not write the hat over the solution $\widehat{T}(M, t)$, the initial condition $\widehat{T}(M, 0)$ and the source function $\widehat{f}(M, t)$, denoting them simply $T(M, t)$, $T(M, 0)$ and $f(M, t)$.

Now we have to get rid of the inhomogeneity in the equation itself, but unlike the procedure for removing the inhomogeneity in the boundary conditions, here we go from the other side: first we solve the homogeneous equation, and then, based on it, we explain how to solve the inhomogeneous one. To see that such a transition is possible, we will rely on Steklov's expansion theorem, since it is precisely what gives the basis in which everything is subsequently expanded. Having solved the homogeneous problem, we obtain the set of its eigenfunctions. Steklov's theorem states that this system is complete: every function with a continuous first and piecewise continuous second derivative, satisfying the same homogeneous boundary conditions, for example (1.16$\displaystyle T(M, t)$ $\displaystyle = 0, \quad M$ $\displaystyle \in S.$) or (1.17$\displaystyle \lambda \cdot \frac{\partial T(M, t)}{\partial \vec{n}}$ $\displaystyle = 0, \quad M$ $\displaystyle \in S.$), expands into a uniformly convergent series in the eigenfunctions of the homogeneous problem (1.9$\displaystyle \frac{\partial T(M, t)}{\partial t}$ $\displaystyle = a^2 \cdot \Delta T(M, t).$). This means that both the sought solution of the inhomogeneous problem and the source function can be represented in the same basis: projecting the inhomogeneous equation onto each eigenfunction and using their orthogonality, we reduce it to independent ordinary equations for the coefficients — that is, we build the solution of the inhomogeneous problem on the solution of the homogeneous one.

We use the method of separation of variables (the Fourier method). We will look for the solution of equation (1.9$\displaystyle \frac{\partial T(M, t)}{\partial t}$ $\displaystyle = a^2 \cdot \Delta T(M, t).$) as a sum of products $T(M, t)$ $= \sum_{n=1}^{\infty} \Psi_n(M) \cdot \phi_n(t)$, where each term is a product of a function of geometry only and a function of time only. The equation is linear and homogeneous, so it suffices to require that each term satisfy it separately; let us consider one product $\Psi(M) \cdot \phi(t)$ and substitute it into equation (1.9$\displaystyle \frac{\partial T(M, t)}{\partial t}$ $\displaystyle = a^2 \cdot \Delta T(M, t).$) — we obtain

which after division by $\Psi(M) \cdot \phi(t)$ leads to

where $\gamma^2$ is a constant. Indeed, the left-hand side of the equality depends only on time, while the right-hand side depends only on the geometry; a function of time can coincide with a function of geometry at all moments of time and at all points of the domain only when both are equal to one and the same constant — this is the essence of the separation of variables. The minus sign in front of it is chosen so that the solution converges, which is easy to verify by examining the equation for time. The specific values of this constant we will determine later — from the equation for the geometry.

Let us write the equation for time and its solution

where $A$ is a constant.

The equation for the geometry has the form

This is the homogeneous Helmholtz equation. To solve it, boundary conditions must be specified. Let us substitute the product $\Psi(M) \cdot \phi(t)$ into the boundary conditions (1.16$\displaystyle T(M, t)$ $\displaystyle = 0, \quad M$ $\displaystyle \in S.$) and (1.17$\displaystyle \lambda \cdot \frac{\partial T(M, t)}{\partial \vec{n}}$ $\displaystyle = 0, \quad M$ $\displaystyle \in S.$) respectively and obtain for the Dirichlet conditions

and for the Neumann conditions

Clearly, the case when $\phi(t)$ $= 0$ is of no interest, since we are constructing conditions for the geometry, not for time. Equation (2.8$\displaystyle \Delta \Psi(M)$ $\displaystyle + \gamma^2 \cdot \Psi(M)$ $\displaystyle = 0.$) with the boundary conditions (2.9$\displaystyle \Psi(M) \cdot \phi(t)$ $\displaystyle = 0$ $\displaystyle \Rightarrow \Psi(M)$ $\displaystyle = 0, \quad M$ $\displaystyle \in S,$) and (2.10$\displaystyle \frac{\partial \Psi(M)}{\partial \vec{n}} \cdot \phi(t)$ $\displaystyle = 0$ $\displaystyle \Rightarrow \frac{\partial \Psi(M)}{\partial \vec{n}}$ $\displaystyle = 0, \quad M$ $\displaystyle \in S.$) is called the Sturm–Liouville problem. It is well studied, and below we will use its known properties without proof (they can be found in the literature). This problem has not one but a countable set of eigenvalues, and each of them corresponds to its own eigenfunction; eigenfunctions corresponding to different eigenvalues are mutually orthogonal and form a complete system.

So, the Sturm–Liouville problem yielded a countable set of eigenvalues $\gamma_n^2$ and the corresponding eigenfunctions $\Psi_n(M)$. For the mode with index $n$ the equation for time gives the factor $e^{- a^2 \cdot \gamma_n^2 \cdot t}$, so that $\Psi_n(M) \cdot e^{- a^2 \cdot \gamma_n^2 \cdot t}$ is a particular solution of equation (1.9$\displaystyle \frac{\partial T(M, t)}{\partial t}$ $\displaystyle = a^2 \cdot \Delta T(M, t).$). Returning to the sum we started with and using linearity, we obtain the general solution

where $c_n$ are arbitrary constants that have absorbed the constant factors of both parts of each term. The completeness of the system of eigenfunctions (Steklov's theorem) guarantees that any admissible solution can be represented by such a sum.

Now we have to determine the coefficients $c_n$; to do so, note that the solution must satisfy the initial condition (2.4$\displaystyle \widehat{T}(M, 0)$ $\displaystyle = T_0(M)$ $\displaystyle - U(M, 0).$), and hence

It is easy to see that the coefficients $c_n$ are determined uniquely thanks to the mutual orthogonality of the eigenfunctions — as the expansion coefficients of the function $T_0(M)$ into a Fourier series in the eigenfunctions $\Psi_n(M)$ — by the general formula (A.4$\displaystyle c_n$ $\displaystyle = \frac{\displaystyle \iiint_G \rho(M) \cdot f(M) \cdot \varphi_n(M) \,dG}{\displaystyle \iiint_G \rho(M) \cdot \varphi_n^2(M) \,dG}.$) from the appendix “Fourier series”

where $\|\Psi_n(M)\|^2$ is the squared norm of the eigenfunction, defined by formula (A.5$\displaystyle \|\varphi_n\|^2$ $\displaystyle = \iiint_G \rho(M) \cdot \varphi_n^2(M) \,dG.$).

Thus, we have obtained the solution of equation (1.9$\displaystyle \frac{\partial T(M, t)}{\partial t}$ $\displaystyle = a^2 \cdot \Delta T(M, t).$) with the initial condition $T_0(M)$ and homogeneous boundary conditions

Now we can proceed to solving the inhomogeneous equation, which, as we said, in accordance with Steklov's expansion theorem, can be built on the solution of the homogeneous problem. To do so, we expand the heat source density function $f(M, t)$ into a Fourier series in the eigenfunctions $\Psi_n(M)$

where $\mu_n(t)$ are the expansion coefficients of the function $f(M, t)$ into a Fourier series in the eigenfunctions $\Psi_n(M)$

We will look for the solution of the inhomogeneous equation in the same form — as a sum of products $\sum_{n=1}^{\infty} \Psi_n(M) \cdot \phi_n(t)$. Let us substitute it together with the expansion (2.14$\displaystyle f(M, t)$ $\displaystyle = \sum_{n=1}^{\infty} \mu_n(t) \cdot \Psi_n(M),$) into the inhomogeneous equation (1.8$\displaystyle \frac{\partial T(M, t)}{\partial t}$ $\displaystyle = a^2 \cdot \Delta T(M, t)$ $\displaystyle + f(M, t),$); since $\Delta \Psi_n(M)$ $= - \gamma_n^2 \cdot \Psi_n(M)$, we obtain

The geometric part remains the same — these are the same eigenfunctions $\Psi_n(M)$. Equating the coefficients of each $\Psi_n(M)$ (this is legitimate due to orthogonality), we see that only the equation for time has changed — it has acquired the term $\mu_n(t)$:

The solution of this equation has the form

where $c_n$ are arbitrary coefficients equal to the coefficients determined in the solution of the homogeneous problem, since they are computed from the initial condition, and at $t$ $= 0$ the integral equals zero.

The solution of the inhomogeneous equation with homogeneous boundary conditions takes the form

One could also obtain the solution of the inhomogeneous equation with inhomogeneous boundary conditions, but this is rather cumbersome — we will construct the final solution later using specific examples.

When solving stationary equations — and the electromagnetic field is generally stationary unless relativistic effects are considered — we need to obtain the general solution without time, and for that we take the limit $t$ $\rightarrow \infty$ in the solution of the nonstationary equation. Note that the first term of the solution vanishes immediately, since the exponential tends to zero and the initial conditions disappear from the solution, which is more than obvious. The heat source density function stops depending on time a priori. It only remains to clarify the integral

Let us write the general solution for the stationary equation

And what if $\gamma_n$ $= 0$? In fact this case is possible, but it is better considered separately for each equation, which is what we will do in the corresponding chapters.