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. 

The problem asks the following: (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.

Solving a system of differential equations whose one of the coefficients is imported data

Suppose we have a coupled system of differential equations: \begin{equation} \frac{db}{dt}=(- \gamma_b -i\omega_b)b-i\frac{g}{2}p;\quad \frac{dp}{dt}=i\frac{g}{2}\Delta N(t) b-(\gamma_a+\gamma_b+2iJ)p. \end{equation} If $ \Delta N$ was fixed, the solution of the system would be like \begin{equation} \begin{pmatrix} b(t)\ p(t) \end{pmatrix}=\begin{pmatrix} a_{11}&a_{12}\ a_{21}&a_{22} \end{pmatrix}\begin{pmatrix} b(0)\ p(0) \end{pmatrix} \end{equation} Using the following code, I have found a $ 2\times 2$ matrix (called sol) whose entries are $ a_{ij}$ in the above equation:

rb=630;wb=75*10^6;g=0.63;ra=2.6*10^6;rm=3.6*10^6;J=6.3*10^7;DeltaN=0.164*10^5; m ={{-rb-I wb,-I g/2},{I g DeltaN/2,-(ra+rm+2 I J)}}; eigvec = Eigenvectors[m] // Transpose // Simplify; eigval = Eigenvalues[m] // Simplify; inv = Inverse[eigvec] // Simplify; v1 = eigval[[1]]; v2 = eigval[[2]]; sol = eigvec.{{E^(v1 t), 0}, {0, E^(v2 t) }}.inv; 

If we suppose that $ p(0)=0$ , then one can easily plot $ |b(t)/b(0)|^2$ : simply plot $ a_{11}(t)$ . But the problem is that $ \Delta N$ is not fixed. It is a $ N\times 1$ matrix which I have obtained from another code written with Fortran and its type is data.txt. The elements of this file are calculated by assuming the time interval between each one is $ 0.001$ . That is, for $ t=0.001$ we have $ \Delta N_1$ , for $ t=0.002$ we have $ \Delta N_2$ , etc. But the time intervals are not included in the txt file.

One way that comes to my mind is this: Assuming we know the analytical form of solfor a fixed $ \Delta N$ , we set time, i.g., equal to $ 0.001$ and then substitute the first row of the txt file (I call it $ \Delta N_1$ ) into sol and find $ a_{11}$ . Then we raise time to $ 0.002$ , substitute $ \Delta N_1$ into sol, find $ a_{11}$ , and repeat the procedure to the last row of the txt file.

Now the question is this: how can I import the txt file to the code and do the procedure that I explained above to get some data like $ \{\{0.001,a11(0.001)\},\{0.002,a11(0.002)\},….\}$ where the first elements are time intervals and the second ones are $ a_{ij}$ corresponding to that particular time?

I had asked a similar question here enter link description here, but in that problem I did not have an external file with txt format.

I could not upload my txt file, so I write the first 10 elements if necessary:

0.164E+05

0.655E+05

0.146E+06

0.258E+06

0.400E+06

0.572E+06

0.776E+06

0.101E+07

0.129E+07

0.159E+07

In what cases is solving Binary Linear Program easy (i.e. **P** complexity)? I’m looking at scheduling problems in particular

In what cases is solving Binary Linear Program easy (i.e. P complexity)?

The reason I’m asking is to understand if I can reformulate a scheduling problem I’m currently working on in such a way to guarantee finding the global optimum within reasonable time, so any advice in that direction is most welcome.

I was under the impression that when solving a scheduling problem, where a variable value of 1 represents that a particular (timeslot x person) pair is part of the schedule, if the result contains non-integers, that means that there exist multiple valid schedules, and the result is a linear combination of such schedules; to obtain a valid integer solution, one simply needs to re-run the algorithm from the current solution, with an additional constraint for one of the real-valued variables equal to either 0 or 1.

Am I mistaken in this understanding? Is there a particular subset of (scheduling) problems where this would be a valid strategy? Any papers / textbook chapter suggestions are most welcome also.

Solving recurrence relation $T(n) \leq \sqrt{n}T(\sqrt{n}) + n$

Given the condition: $ T(O(1)) = O(1)$ and $ T(n) \leq \sqrt{n}T(\sqrt{n}) + n$ . I need to solve this recurrence relation. The hardest part for me is the number of subproblems $ \sqrt{n}$ is not a constant, it’s really difficult to apply tree method and master theorem here. Any hint? My thought is that let $ c = \sqrt{n}$ such that $ c^2 = n$ so we have $ T(c^2) \leq cT(c) + c^2$ but I does not look good.

Solving shortest path problem with Dijkstra’s algorithm for n negative-weight edges and no negative-weight cycle

Although many texts state Dijkstra’s algorithm does not work for negative-weight edges, the modification of Dijkstra’s algorithm can. Here is the algorithm to solve a single negative-weight edge without negative-weight edges.

Let $ d_s(v)$ be the shortest distance from source vertex s to vertex v.
Suppose the negative edge $ e$ is $ (u, v)$
First, remove the negative edge $ e$ , and run Dijkstra from the source vertex s.
Then, check if $ d_s(u) + w(u, v) \leq d_s(v)$ . If not, we are done. If yes, then run Dijkstra from $ v$ , with the negative edge still removed.
Then, $ \forall t \in V $ , $ d(t) = min(d_s(t), d_s(u) + w(u, v) + d_v(t))$

Given the above algorithm, I want to modify the above algorithm again to solve n negative-weight edges and no negative weight cycle. Any hint?

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]}]; 

Is “Solving two-variable quadratic polynomials over the Integers” is an NP-Complete Problem?

On this Wikipedia article, it claims that given $ A, B, C \geq 0, \; \in \mathbb{Z}$ , finding $ x, \,y \geq 0, \, \in \mathbb{Z}$ for $ Ax^2+Bx^2-C=0$ is NP-complete? Given by how easy I can solve some (with nothing but Wolfram), it doesn’t seem right. I’m sure it’s either written incorrectly or I’m just misunderstanding something.

What are the correct steps in solving polygon monotone triangulation?

I am working out enter image description here step by step and I am stuck on vertex 7. I got that it was a regular vertex and helper(e_i-1) is not a merge vertex so I look for the leftmost edge in the sweep line. My question is, would e6 be considered to the left of it, or is it none? Any already completed examples that I could see would help me understand this greatly.