## 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 particular case of Bernoulli Equation

I have a Bernoulli equation (attached below). So this is need to be converted to ODE. I tried to solve it by a separation of variables but I could not. Can anyone help me with the code for this, please?

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

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

## What captcha solving services you recommend?

GSA CB + what else people are using these days?