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My matrix was : [ [0,1,2,3,1], [2,0,4,2,4], [1,2,0,4,1], [4,3,2,0,3], [3,4,3,4,0] ] Team 5: rating = 0.5552749621639953 Team 2: rating = 0.493080305685833 Team 4: rating = 0.4796470095274982 Team 3: rating = 0.3560934619397378 Team 1: rating = 0.3027832907437918 Team 5 was the best.
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Fixed it using Overleaf, made it easy to see places where I had typed [ instead of {.
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The full Fourier Series is generally: $$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} a_ncos(\frac{n\pi x}{L}) + b_nsin(\frac{n\pi x}{L})$$ First find the value of $a_0$ for $f(x) = x - x^2$ over the interval $[0,L]$ $$a_0 = \frac{2}{L} \int_{0}^{L} f(x)\,dx$$ $$\implies a_0 = 4 \int_{0}^{\frac{1}{2}}(x - x^2)\,dx =…
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I am having trouble figuring out why the these remaining lines won't compile. I have used most of the editing window to fix other things, just can't seem to find what's wrong.
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The full Fourier series is generally: $$ f(x)~ \frac{a_0}{2} + \sum_{n=1}^{\infty} a_ncos(\frac{n\pi x}{L}) + b_nsin(\frac{n\pi x}{L})$$ First find the value of $a_0$ for $f(x) = x - x^2$ over the interval $[0,\frac{1}{2}]$ $$ a_0 = \frac{2}{L} \int_{0}^{L} f(x)\,dx $$ $\implies a_0 = 4 \int_{0}^{\frac{1}{2}}(x - x^2)\,dx$…
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For my unique conditions, $\kappa = 0$ and $f = 0$. The three sides on the lower left, sticking out of the bar, are set to temperature zero. The side at the very top of the rectangle on the upper right is set to temperature 1. The remaining sides are insulated. At the steady state for the heat distribution, the temperature…
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Here's the photo of my house at t=0.6, the temperature just above the door is 0.37670. I had set the diffusivity constant equal to 1.
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For my problem, the basic heat equation was: $$u_t = 0.47\Delta u$$ The vertical side of the triangle is insulated The Dirichlet condition on the circle and the 2 other sides of the triangle is $$u(x,y,t) = 0.87^2 - (x^2+y^2)$$ The initial condition is: $$u(x,y,0) = 0.87^2 - (x^2+y^2)$$ I found the temperature of the…
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The initial value problem is $$u_t + u_x = 3u, u(x,0) = x^2 .$$ General form of 1st order, liner PDE is: $$u_t + cu_x + au = f(x,t).$$ Therefore adjusting the initial problem gives: $$u_t + u_x - 3u = 0.$$ With $c = 1, a = -3$ and $f(x,t) = 0$ To solve this question, we first need perform a change of variables with $\chi =…
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Start with the equation: $$u_t = k(t)u_{xx} .$$ Substitute $\tau = \int_0^t k(\eta) d\eta$ in for $t$. Solving the integral gives: $$\tau = K(t) - K(0).$$ After the substitution, $u(x,t)$ becomes $u(x,\tau).$ Since the function to find $\tau$ does not contain any $x$ terms, $u_{xx}$ is unaffected by the substitution:…
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The basic conservation law expressed as a PDE is: $$u_t + \phi_x = f.$$ To find Burgers equation, first assume that the source $f = 0$ and the flux $\phi = \frac{1}{2}u^2$. To find $\phi_x$ take the chain rule derivative of $\frac{1}{2} * u(x,t)^2$ with respect to $x$. Therefore $\phi_x = u(x,t) * u_x(x,t)$ Inputting back…
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A metal bar of length 1 lies along the unit interval. Its temperature distribution is given by: $$g(x) = 5x^2 - 1x .$$ At time t = 0, its left end is set to temperature -2 and its right end to -1. Sketch the temperature distribution at times: t = 0, t = 0.01, t = 0.1, t = 10.
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Heat flow with a constant internal heat source is governed by: $$u_t = 1u_{xx} + 2, u(0,t) = -3, u(1,t) = 5.$$ When at the steady state temperature distribution, $u_t = 0$ therefore: $$0 = u_{xx} + 2$$ This can be rearranged to give $u_{xx} = -2$. Next this function was integrated with respect to $x$ twice giving $$u(x) =…
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The displacement from the equilibrium of the midpoint of the string at time t=1.3 seconds into the vibration is approximately -0.351.