# Solving eigenvalue BVP with an interface

I have a boundary-value problem, that is defined over two adjacent regions with an interface in the middle, that contains an eigenvalue $\lambda$. The boundary conditions and the equations are homogeneous (as expected a linear stability analysis), but could depend on $x$.

For a simple example: \begin{align} y''''(x) + 5 y''(x) + \lambda^4 y(x) &= 0, \quad x \in [x_1,x_2] \\ z''''(x) - \lambda^4 z(x) &=0, \quad x \in [x_2,x_3] \\ \end{align}

With some boundary conditions, say \begin{align} y'(x_1)= y''(x_1)=0, \quad z'(x_3)=z'''(x_3)=0, \end{align} and some continuity/jump conditions at the interface: \begin{align} y(x_2)&=z(x_2) \\ y''(x_2)&=z''(x_2) \\ y'(x_2)+y'''(x_2)&= - 2 z'(x_2)+z(x_2)-z'''(x_2) \\ 3 y'''(x_2)&= z'''(x_2)-z'(x_2) \end{align}

Here is some code with those equations and conditions:

x1 = -5; x2 = 1; x3 = 2;
eq1 = y''''[x] + 5 y''[x] + λ^4 y[x] == 0;
eq2 = z''''[x] - λ^4 z[x] == 0;
matchconds = {y[x2] == z[x2],  y'[x2] + y'''[x2] == -2 z'[x2] + z[x2] - z'''[x2],
y''[x2] == z''[x2], 3 y'''[x2] == -z'[x2] + z'''[x2]};
bcs1 = {y'[x1] == 0, y''[x1] == 0};
bcs2 = {z'[x3] == 0, z'''[x3] == 0};


These equations are actually somewhat amenable to finding analytic results, which may be useful to compare fully numerical solutions with, but in general the coefficients of $y$ and $z$ can depend on $x$. Here is a code for finding some roots via DSolve:

ysub = DSolve[eq1, y, x][[1]];
zsub = DSolve[eq2, z, x, GeneratedParameters -> (C[# + 4] &)][[1]];
coefmat = Transpose[Table[Coefficient[Join[bcs1, bcs2, matchconds] /. ysub /. zsub /.
Equal -> Subtract, ii], {ii, Array[C, 8]}]];
detRoots = {λ, 0} /. (FindRoot[Det[coefmat], {λ, #}] & /@ {1.3, 1.5, 2, 4}) //Chop;


Note, I plan on self-answering this using my package which calculates the Evans function, but I'm interested in other methods.

• What is your question? Are you looking for all lamda (detRoots) in a given range? Aug 14, 2018 at 18:32
• @UlrichNeumann Yes, say find all the values of $\lambda \leq 10$. As I say in the question, I have a method (and package) for doing this myself, but I'm interested in what other approaches exist. Aug 14, 2018 at 19:53

In general, for a linear homogeneous equation of this type we can write: \begin{align} \frac{d \mathbf{y}}{dx} &= \mathbf{A}_y(\lambda, x) \cdot \mathbf{y}, \\ \frac{d \mathbf{z}}{dx} &= \mathbf{A}_z(\lambda, x) \cdot \mathbf{z}, \\ \mathbf{B} \cdot \mathbf{y}(x_1) &= \mathbf{0}, \\ \mathbf{F} \cdot \mathbf{y}(x_2) +\mathbf{G} \cdot\mathbf{z}(x_2) &= \mathbf{0}, \\ \mathbf{C} \cdot \mathbf{z}(x_3) &= \mathbf{0}. \\ \end{align}

for some matrices $$\mathbf{A}_y, \mathbf{A}_z, \mathbf{B}, \mathbf{C}, \mathbf{F}, \mathbf{G}$$, which may all involve the eigenvalue $$\lambda$$ (and $$\mathbf{A}_y$$ and $$\mathbf{A}_z$$ may be functions of $$x$$).

The boundary condition at $$x=x_1$$ gives us two conditions on the four entries of $$\mathbf{y}$$ there, so we can find two linearly independent solutions that satisfy the boundary conditions. In this example, $$\mathbf{y}^1(x_1) = [1, 0, 0, 0]$$ and $$\mathbf{y}^2(x_1) = [0, 0, 0, 1]$$. We can then integrate both of these solutions across to the matching point $$x_2$$, and then the general solution is given by $$\mathbf{y} = k_1 \mathbf{y}^1 + k_2 \mathbf{y}^2$$ (due to linearity).

The same procedure starting at $$x=x_3$$ gives us a general solution for $$\mathbf{z} = k_3 \mathbf{z}^1 + k_4 \mathbf{z}^2$$.

For the simpler case without the interface conditions (where $$\mathbf{A}_y = \mathbf{A}_z$$, we would then require that the two solutions match at a matching point (which could be chosen arbitrarily), i.e. $$\mathbf{y}(x_m) = \mathbf{z}(x_m)$$, which could be written as $$\mathbf{N}(x_m, \lambda) \mathbf{k}=\mathbf{0}$$, where the matrix $$\mathbf{N}$$ is given by $$$$\mathbf{N}(x_m, \lambda)=[\mathbf{y}^1, \mathbf{y}^2, \mathbf{z}^1, \mathbf{z}^2].$$$$ Non-trivial solutions (i.e. eigenvalues) therefore require $$|\mathbf{N}(x_m, \lambda)|=0$$. The Evans function $$D(\lambda)$$ is a (complex) analytic function whose roots are eigenvalues of the original equation, which is independent of the location of the matching point, $$$$D(\lambda)= \exp( -\int^{x_m}_{x_1} {\rm tr}\, A(s, \lambda)\, d s) \; |N(x_m, \lambda)|.$$$$

For the interface case as described here, we need to satisfy the interface conditions instead, leading to $$$$\hat{\mathbf{N}} = [\mathbf{F} \cdot \mathbf{y}^{1}, \mathbf{F} \cdot \mathbf{y}^{2}, \mathbf{G} \cdot \mathbf{z}^{1}, \mathbf{G} \cdot \mathbf{y}^{2}],$$$$ and therefore $$|\hat{\mathbf{N}}|=0$$.

I have a package that implements all this, including using the compound matrix method to help make the differential equations less stiff, at the cost of converting to more equations. So let us load the package:

Needs["PacletManager"]
PacletInstall["CompoundMatrixMethod",
"Site" -> "http://raw.githubusercontent.com/paclets/Repository/master"]
Needs["CompoundMatrixMethod"]


Convert the system of ODEs into the matrix form, which gives all the matrices:

sys = ToMatrixSystem[{eq1, eq2}, {bcs1, bcs2, matchconds}, {y,z}, {x, x1, x2, x3}, λ];


Then the function Evans can be evaluated for a given value of $$\lambda$$ with this system:

Evans[1, sys]
-0.170854


And a quick plot:

Plot[Evans[λ, sys], {λ, 0, 5}]


Note that even though there is a zero of the analytic determinant at $$\lambda = 1.58114$$, this is because there are repeated roots of the equation for $$y$$ and not an actual eigenvalue. Note that the Evans function is continuous here (goes down to ~-75).

And then we can find the eigenvalues via FindRoot:

λ /. FindRoot[Evans[λ, sys], {λ, #}] & /@ {1, 1.3, 1.4, 5}

• if a problem has two eigenvalues represented by two variables, does this package can handle it using ToMatrixSystem[{eq1, eq2}, {bcs1, bcs2, ...}, {y,z}, {x, x1, ...}, {λ1, λ2}]? Jul 8, 2020 at 12:38
• @Nobody, no, I haven't done anything with systems where there are two eigenvalues, I'm unaware of what they might look like - feel free to show me an example. If you just have a complex eigenvalue that you are splitting up, you can use that directly in my method. Jul 8, 2020 at 18:32
• Hi,@SPPearce, for the interface case, we integrate both from $x_1$ to $x_2$ and $x_3$ to $x_2$. In this case, what are the integral limits in the definition of $D(\lambda)$? Should it be $\int_{x_1}^{x_2}$ or $\int_{x_3}^{x_2}$? Or, whether we have two $D$ functions here? Thank you in advance! Aug 5, 2023 at 6:03
• Hi Nobody, Been a while since I've looked at this, but I think you can integrate from either end, the zeroes should coincide. Aug 11, 2023 at 5:43

An easy way to approximate the \[Lambda] in the range 0 < \[Lambda] < 10 is to use plot and its mesh functionality:

detM = Det[coefmat];
pict = Normal[Plot[detM , {\[Lambda], 0, 10}, PlotRange -> {-1, 1},
MeshFunctions -> {#2 &}, Mesh -> {{0}}, PlotPoints -> {100, {0}}, WorkingPrecision -> 20, AxesLabel -> {"\[Lambda]", "detM"}]]


The plot shows 6 roots

p0 = Sort [Cases[ pict[[All, 1]], _Point, Infinity] /. Point -> Identity]
\[Lambda]i = Select[p0, #[[2]] < 0.001 &] [[;; , 1]]
(*{0., 1.0101*10^-7, 0.959649, 1.33247, 1.48215, 4.54458, 7.79846}*)


The \[Lambda]i could be used as starting values for FindRoot to improve the approximation...

• Sure, although I'm interested in methods that don't require an analytic solution to the individual differential equations in the two domains to exist. Aug 15, 2018 at 9:46