I'm interested in solving a heat equation with a discontinuous source term in one dimension on the real line: ∂ u ∂ t ( x , t ) = − x θ ( x ) u ( x , t ) − x θ ( − x ) u ( − x , t ) + D ∂ 2 u ∂ x 2 ( x , t )where θ ( ⋅ ) is the Heaviside function. The boundary conditions are that lim | x | → ∞ u ( x , t ) = 0 The initial condition is arbitrary, but even for a symmetric Gaussian or a symmetric exponential an analytical solution would be interesting.Because of the discontinuity in the right-hand side induced by θ ( ⋅ ), my intuition would be to solve in the two domains x&gt;0 and x&lt;0 and patch them together at x=0. However, the source terms have the effect of causing exponential decay with rate x at the point x for x&gt;0 and corresponding growth at the point −x. This introduces an asymmetry, which should cause a diffusive flux back to the side x&gt;0. Solving for the solution in the two domains does not seem to capture this feedback, and I'm not sure how to solve globally. What are some avenues to proceed?

Question

I&#039;m interested in solving a heat equation with a discontinuous source term in one dimension on the real line:            ∂      u              ∂      t        (  x  ,  t  )  =  −  x  θ  (  x  )  u  (  x  ,  t  )  −  x  θ  (  −  x  )  u  (  −  x  ,  t  )  +  D                    ∂        2            u              ∂              x        2              (  x  ,  t  )where   θ  (  ⋅  ) is the Heaviside function. The boundary conditions are that       lim                  |            x              |            →      ∞        u  (  x  ,  t  )  =  0 The initial condition is arbitrary, but even for a symmetric Gaussian or a symmetric exponential an analytical solution would be interesting.Because of the discontinuity in the right-hand side induced by   θ  (  ⋅  ), my intuition would be to solve in the two domains x&amp;gt;0 and x&amp;lt;0 and patch them together at x=0. However, the source terms have the effect of causing exponential decay with rate x at the point x for x&amp;gt;0 and corresponding growth at the point −x. This introduces an asymmetry, which should cause a diffusive flux back to the side x&amp;gt;0. Solving for the solution in the two domains does not seem to capture this feedback, and I&#039;m not sure how to solve globally. What are some avenues to proceed?

Samsonitew7b · Accepted Answer

As I understand your equation, it can be written as      u    t    =  −  x  u  (      |    x      |    ,  t  )  +      u          x      x        .Take first the case where x&amp;gt;0. A separation of variables   u  (  x  ,  t  )  =      e          −      c      t        X  (  x  ) gives  −  c  =  −  x  +            X      ″        X    .A bounded solution to this uses the Airy function Ai,  X  (  x  )  =      A    i    (  x  −  c  )  ,because             A      i        ″    (  x  )  =  x        A    i    (  x  ) Take c to be minus a negative root of Ai; there are infinitely many choices.So we have  u  (  x  ,  t  )  =      e          −      c      t            A    i    (  x  −  c  )  ,    (  x  &amp;gt;  0  )  .Then for x&amp;lt;0 we need to solve      u    t    =  −  x      e          −      c      t            A    i    (  −  x  −  c  )  +      u          x      x        .For this it is sufficient to take  u  (  x  ,  t  )  =  −      e          −      c      t            A    i    (  −  x  −  c  )  ,    (  x  &amp;lt;  0  )  .Thus u is an odd function of x, and with the choice of c, u is continuous and once differentiable with respect to x at x=0.To get existence of       u          x      x        (  0  ,  t  ) this can be achieved by a linear combination of two such functions with different values of c, arranged so that the second derivative is zero at the origin.

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