In electrochemistry, electrolyte movement towards electrode is monitored by time and distance variable. How can I solve such a partial differential equation of two variables with Laplace transformation?
Transform the equation with respect to both time and space. For example, suppose you have a one dimensional diffusion equation
\frac{\partial n(x,t)}{\partial t} = D\partial^2 n(x,t)}{\partial x^2}
Transforming it leads to
(s+Dk^2)n(k,s) = initial + boundary conditions.
Solve this for n(k,s) which can then be back transformed to n(x,t)
BTW, wouldn't it be nice if ResearchGate supported mathjax or something similar?
Transform the equation with respect to both time and space. For example, suppose you have a one dimensional diffusion equation
\frac{\partial n(x,t)}{\partial t} = D\partial^2 n(x,t)}{\partial x^2}
Transforming it leads to
(s+Dk^2)n(k,s) = initial + boundary conditions.
Solve this for n(k,s) which can then be back transformed to n(x,t)
BTW, wouldn't it be nice if ResearchGate supported mathjax or something similar?
Laplace and Fourier transforms essentially turn differential equations into algebraic equations.
In my experience, the biggest problem with Laplace transforms is the inversion - it's not always that easy to transform back to the (x,t) domain. I assume the PDE is first-order in time and second-order in space. The Laplace transform is tailor made to initial value problems (in fact, I don't see a straight forward way to implement boundary conditions), so just apply the transform to the time component. This will give you a second order ODE in space, which you can hopefully solve analytically. Invert the solution to get the time-dependent result.
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