Show that reaction-diffusion PDE is linear

[axeeonn]

du/dt = f(u)+D(laplacian*u)

where u=u(x,y,t), D>0, f(u)=a*u

I'm not sure where to start other than substitute f(u) with a*u. Suggestions?

 

[arildno]

Rewrite your equation in the form:

L(u)=0, L=d/dt-a-D*Laplacian.
Here, L is called a differential operator that works on the function u.

To show that L is linear, you must show that for any functions U,u , and constants c, b that:
L(c*U+b*u)=c*L(U)+b*L(u)

We have:
L(c*U+b*u)=d/dt(c*U+b*u)-a*(c*U+b*u)+D*Laplacian(c*U+b*u)=
=c*(dU/dt-a*U+D*laplacian(U))+b*(dU/dt-a*U+D*laplacian(U))=
c*L(U)+b*L(u),
as required.
Note that if U and u are both solutions to the original differential equation;
i.e. L(U)=L(u)=0,
the linearity of L implies that any sum of U and u also is a solution.

 

[M98Ranger]

To show that the reaction-diffusion PDE is linear, we need to prove that it satisfies the properties of linearity, which are superposition and homogeneity.

First, let's consider the property of superposition. This means that if we have two solutions u1 and u2 for the PDE, then the sum of these solutions, u1 + u2, will also be a solution. So, let's assume that u1 and u2 are solutions for the PDE:

du1/dt = f(u1) + D(laplacian*u1)
du2/dt = f(u2) + D(laplacian*u2)

Now, let's take the sum of these two equations:

du1/dt + du2/dt = f(u1) + f(u2) + D(laplacian*u1) + D(laplacian*u2)
= f(u1 + u2) + D(laplacian*(u1 + u2))

Since f(u) = a*u, we can rewrite the above equation as:

du1/dt + du2/dt = a*(u1 + u2) + D(laplacian*(u1 + u2))
= a*u + a*u + D(laplacian*u) + D(laplacian*u)
= a*u + D(laplacian*u) + a*u + D(laplacian*u)
= du/dt + du/dt

Therefore, u1 + u2 is also a solution for the PDE, satisfying the property of superposition.

Next, let's consider the property of homogeneity. This means that if we have a solution u for the PDE, then any constant multiple of u, c*u, will also be a solution. So, let's assume that u is a solution for the PDE:

du/dt = f(u) + D(laplacian*u)

Now, let's multiply both sides of the equation by a constant c:

c*du/dt = c*f(u) + c*D(laplacian*u)

Since f(u) = a*u, we can rewrite the above equation as:

c*du/dt = a*(c*u) + D(laplacian*(c*u))
= f(c*u) + D(laplacian*(c*u))

Therefore, c*u is also a solution for the PDE,