## I need your help guys wo solve inverse laplace questions

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## Solving Laplace equation inside and outside a sphere using NDSolve in Mathematica?

Let us consider we have a hollow sphere of of radius R.The surface of the potential is at some fixed potential.I want to solve the Laplace equation for both inside and outside the sphere.To make the problem little bit more complicated let’s choose that the inside and outside medium of the sphere have different permittivity.Most of us know how to solve the problem analytically.
Now how can I solve this problem in Mathematica using NDSolve?

## Want to solve Laplace equation for two mediums separated by an interface

I want to solve the Laplace equation for electrostatics in two different regions separated by an interface. I can’t understand how to impose a continuity condition for the electric field on the interface. Can anyone help me to figure out the problem?

## How can I evaluate this laplace transform?

begin{align}mathcal Lleft{t^{n-1}cotleft(frac t2right)sin(omega t)right}(s)=,?end{align}

## laplace transform – Hello guys, kindly check my solution and my the other solution why the values of B are different ?!

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## Solving the heat equation using Laplace Transforms

I am trying to solve the 1-D heat equation using Laplace Transform theory. The equation is as follows. I don’t have the capability to write the symbols so I will write it out.

                     partial u/partial t = 2(partial squared u/ partial x squared) -x
boundary conditions are partial u/partial x(0,t)=1, partial u/partial x(2,t)=beta.


(a). For what value of beta does there exist a steady-state solution?
(b). if the initial temperature is uniform such that u(x,0)=5 and beta takes the value suggested by the answer to part (a), derive the equilibrium temperature distribution.

I was able to get an equation that looks like U(x,s)=c e^(s/2)^1/2 -(1/s)((x/s)-u(x,0)). But I am not sure how to go from here to solve for beta using the boundary conditions. I need some assistance from someone.

## fa.functional analysis – Is the indicator function on (0,1) the laplace transform of some function?

Let $$F(s)=chi_{(0,1)}(s)$$ be the indicator function on the interval (0,1). I am wondering if there exists a function or distribution $$f(t)$$ such that
$$F(s)=int_0^infty f(t)e^{-st}dt.$$

I have tried to use the inverse Laplace transform to compute $$f(t)$$ (at least formally), but it seems difficult. I am still not sure about the existence of $$f(t)$$.

## differential equations – Solving Laplace PDE with DSolve

I’m trying to get an analytical solution of Laplace PDE with Dirichlet boundary conditions (in polar coordinates). I managed to solve it numerically with NDSolveValue and I know there is an analytical solution and I know what it is, but I would like DSolve to return it. But DSolve returns the input.

sol = DSolve({Laplacian(
u((Rho), (CurlyPhi)), {(Rho), (CurlyPhi)}, "Polar") == 0,
DirichletCondition(u((Rho), (CurlyPhi)) == 0,
1 <= (Rho) <= 2 && (CurlyPhi) == 0),
DirichletCondition(u((Rho), (CurlyPhi)) == 0,
1 <= (Rho) <= 2 && (CurlyPhi) == (Pi)),
DirichletCondition(
u((Rho), (CurlyPhi)) == Sin((CurlyPhi)), (Rho) == 1 &&
0 <= (CurlyPhi) <= (Pi)),
DirichletCondition(
u((Rho), (CurlyPhi)) == 0., (Rho) == 2 &&
0 <= (CurlyPhi) <= (Pi))},
u, {(Rho), 1, 2}, {(CurlyPhi), 0, (Pi)});


## laplacian – maximum principle, Inhomogeneous Laplace equation

I am working on PDE problem given by my teacher.

$$Omega$$ is a bounded domain. Function $$u$$ is a solution of the following equation:

$$Delta u -mu^2 u = f$$ in $$Omega$$

$$u=0$$ on $$partial Omega$$

Show that $$max_Omega |u| leq 1/mu^2max_Omega |f|$$.

I tried to apply a maximum principle but don’t know how to use it for inhomogeneous case with $$u$$-term on the RHS.
My another attempt is:

$$max_Omega |u| = max_Omega |(Delta u – f)/ mu^2| leq 1/mu^2 max_Omega |Delta u|+ 1/mu^2max_Omega |f|$$

But then I am getting stuck.

Any help will be greatly appticiated.

## Laplace transform of absolute value function

I’m trying to compute the Laplace transform of the following function:
$$f(x)= frac{ |x-a| }{x-a}$$
with $$a>0$$.

I would like to know if I’m right. For $$x>a$$, $$f(x)=1$$ and for $$x, $$f(x)=-1$$. So, $$mathcal{L}left{ f(x)right} = left{ begin{matrix} -frac{1}{s}, & mbox{ 0a} end{matrix} right.$$