Graph of Laplace equation

I have solved the following Laplace equation

a0 = (1/Pi) Integrate[Cos[φ]^2 + Sin[φ]^3, {φ, 0, 2 Pi}] an = (1/Pi) Integrate[(Cos[φ]^2 + Sin[φ]^3)*Cos[n*φ], {φ, 0, 2 Pi}] Plot[an, {n, 0, 10}] bn = (1/Pi) Integrate[(Cos[φ]^2 + Sin[φ]^3)*Sin[n*φ], {φ, 0, 2 Pi}] ann = an*Cos[n*φ] bnn = bn*Sin[n*φ] a = Sum[((r/4)^n)*(ann + bnn), {n, 1, Infinity}] f[r_, φ_] := a0/2 + a ParametricPlot3D[{r,φ, f[r, φ]}, {r, 0, 1}, {φ, 0, 2 Pi}] Plot3D[f[r, φ], {r, 0, 1}, {φ, 0, 2 Pi}] 

My problem is that Plot3D and ParametricPlot3D doesn’t work. Any help?

finding solutions numerically of the equation

given the equation x^5 – y^5 = 1 ,x-2y=1 i need to find the numerical values of the solutions. Now as far as i know N is the command that we’ll be using to get numerical values , NSolve for solving the equation numerically and that it must contain in the output, numbers , not symbols, but i havent been specified either to solve for x or y, nor have i been given values for any of them . if i choose to solve for x , the output gives solution containing y and solving for y gives result in x, these arent numerical forms of solutions. the N command is only used to give a value numerically. how can i get that in this equation? or maybe i can use another command that can be helpful?

Inhomogeneous Heat Equation Formula Not Satisfying IVP

From Wikipedia,


However, when I plug the formula of $ u(x,t)$ into Mathematica 12, it doesn’t seem to satisfy the PDE (does not give $ 0$ ):

enter image description here

Here is the code:

G[x_, t_] := 1/Sqrt[4*Pi*k*t]*Exp[-x^2/(4*k*t)]  u[x_, t_] := Integrate[G[x - y, t - s]*f[y, s], {y, -Infinity, Infinity}, {s, 0, t}]  FullSimplify[D[u[x, t], t] - k*D[u[x, t], {x, 2}] - f[x, t], Assumptions -> t > 0] 

Why does renderig my equation in Mathematica using MaTeX does not work out?

I have some problem with showing this LaTeX input:

$    F = (x_1  \lor x_2)  \land (x_1 \lor \overline x_3) \land (x_2 \lor x_4) \land (\overline x_3 \lor \overline x_4) \land (\overline x_1 \lor \overline x_4) $   

I tried using MaTex["F = (x_1 \lor x_2) \land (x_1 \lor \overline x_3) \land (x_2 \lor x_4) \land (\overline x_3 \lor \overline x_4) \land (\overline x_1 \lor \overline x_4)"] but it does not work.

Do we have any other solution to show this text in Matematica?

Fitting plot and data to an equation

If I have the following data:

data={{2, 66.7635}, {Log[300]/Log[10], 69.9679}, {Log[600]/Log[10],    71.54}, {3, 72.2428}, {-2.30103, 54.0023}, {-(Log[60]/Log[10]),    55.1941}, {-(Log[20]/Log[10]), 56.0038}, {-1,    56.9497}, {-(Log[6]/Log[10]), 57.305}, {-(Log[10/3]/Log[10]),    57.7213}, {-(Log[2]/Log[10]), 58.2489}, {-2.30103,    54.0367}, {-(Log[60]/Log[10]), 55.1157}, {-(Log[20]/Log[10]),    56.1704}, {-1, 56.7117}, {-(Log[6]/Log[10]),    57.2506}, {-(Log[10/3]/Log[10]), 57.7097}, {-(Log[2]/Log[10]),    58.1068}} 

Which looks like this plotted:

enter image description here

I have two questions:

1) How can I fit and plot the fit of this data based on the following equation?

: `` where Tf'_ref=57.2506 , q_ref=0.166667 and c1 and c2 are the fitting parameters. Also, notice that data is Tf' vs Log q in the equation.

2) How can I find the values of c1 and c2 which are the fitting parameters.

The fitting (orange line) is supposed to look like this (done in excel):

enter image description here

EDIT: I tried using NonLinearFitModel like this: Table[{NonlinearModelFit[data, Logqref - ((c1*(data[[i, 2]] - Tfref))/(c2*(data[[i, 2]] - Tfref))), {{c1, 8.6}, {c2, 17.2}}, x]; }, {i, 1, 11}] but this does not work. The reason I tried this is because data[[i, 2]] represents Tf' in the equation. Here Logref=Log10[0.16667]

Is solving a quadratic equation using Turing machine impossible?

I’ve just started Algorithms at university. There’s a task to write an algorithm for a Turing machine to solve quadratic equations. The task doesn’t specify if it’s x^2+bx+c or ax^2+bx+c. I’ve searched whole bunch of information over Russian and English Internet.

I did find articles, which say it’s not possible because we’ve got real numbers A, B, C. Please confirm if that’s true. I may not get it correct.. But I think that’s impossible. I still don’t know how to prove my thoughts.

Thanks in advance!

1D Wave Equation: Vertical Rod and Displacement vs. Textbook Solution

I am trying to setup Mathematica to analyze a vertical round rod under its own weight, fixed on one end free on the other. I have the 1D wave equation and a distributed load to represent the self weight of the round rod.

Vertical Rod Layout

The problem is when I compare the Mathematica solution to the textbook solution the two do not agree.

Sample problem is given below.

Y = 199*^9; (*Young's modulus in Pa *) \[Rho] = 7860; (* Steel density in kg/m^3*) dia = 1/39.37; (* 1" dia converted to meters*) c = Sqrt[Y/\[Rho]]; len = 1000; (*length in meters*) tmax = 5; (* Max time for analysis*) area = \[Pi]*dia^2/4; (*Round rod cross sectional area*) wtfactor = \[Rho]*9.81*area/len;  frwt[x_] := \[Rho]*    area*9.81*(1 -       x/len); (*Rod Self weight imposed as a distributed load*) nsol6 = NDSolve[{\!\( \*SubscriptBox[\(\[PartialD]\), \({t, 2}\)]\(z[x, t]\)\) == c^2*\!\( \*SubscriptBox[\(\[PartialD]\), \({x, 2}\)]\(z[x, t]\)\) + frwt[x] +       NeumannValue[0, x == len],    z[0, t] == 0}, z[x, t], {x, 0, len}, {t, 0, tmax},   Method -> {"FiniteElement",      "MeshOptions" -> {"MaxCellMeasure" -> 10}}   ] fnnsol6[x_, t_] = nsol6[[1, 1, 2]] Plot3D[fnnsol6[x, t], {x, 0, len}, {t, 0, tmax},   PlotLabels -> Automatic, AxesLabel -> Automatic]  deltaL = ((\[Rho]*9.81*len^2)/(  Y*2)) (*Textbook elongation for a vertical rod under self weight*) calcdeltaL =   fnnsol6[len,    5] (*Calculated delta Length from PDE solution.  Should match \ textbook*)  deltaLfunc[x_, l_] := \[Rho]*9.81*   x*(2*len - x)/(2*Y) (*Verified Correct*) xydata = Thread[{Range[0, 1000, 100],      deltaLfunc[x, 1] /. {x -> Range[0, 1000, 100]}}]; xydata2 =   Thread[{Range[0, 1000, 100],     Reverse[a]}]; (*Same answer different calc format for debugging*) Show[Plot[fnnsol6[x, 0], {x, 0, len}, PlotLabels -> {"PDE Val"},    PlotRange -> All   ],  ListLinePlot[xydata2, PlotStyle -> Green, PlotLabels -> {"Correct"}]] 

If you’ve read this far, thank you.

In summary my question is: Is this a Mathematica issue or a PDE setup problem? The PDE is right out of a textbook so I don’t think that’s the problem but Mathematica gives no errors and I am out of troubleshooting ideas so looking for some help.

Thank You

how to substitute machine numbers into an equation

I have a system of equations in which variables are indexed as:

8 x[1] + 2 y[1] == 2; 3 x[1] - 5 y[1] == 7; 

The solution obtained from my model is of the form:

sol = {{x[1.] -> 12/23, y[1.] -> -(25/23)}}; 

As seen, the variable index looks like a Machine Number (x[1.] and y[1.]). Therefore, I cannot map the sol onto the equations to check if they are satisfied.

I simply want the sol to be:

{{x[1] -> 12/23, y[1] -> -(25/23)}}; 

How can I get rid of the Machine Numbers to be the variable index?

Is grammar that describes an equation in prefix (Polish) notation always unambiguous?

I recently completed a problem in which I was asked to generate a parse tree for the expression $ + \, 5 \, * \, 4 \, 3$ using the following grammar and rightmost derivation:

$ $ Expr \rightarrow + \, Expr \, Expr \, | \, * \, Expr \, Expr \, | \, 0 \, | \, \dots \, | \, 9 \,$ $

While I have no trouble taking the derivation and creating its parse tree, the question also asks whether the grammar is ambiguous. In the scope of what I’ve been taught, my only tool for proving ambiguity has been to find a different parse tree for whatever leftmost or rightmost derivation I have, thus proving multiple valid parses and ambiguity. However, I have not been told how to prove unambiguity. I am fairly confident that the grammar described above is unambiguous based partially on intuition, and partially because it’s designed for prefix notation. I tried to generate new trees for a given string to prove ambiguity, but since the operator is always leftmost, I could not find any string in which multiple parse trees could be created. If I am mistaken, please let me know.

My question is this: Is it possible for grammar that describes strings using prefix (Polish) notation such as the one above to ever be ambiguous? My intuition tells me that it will always be unambiguous, but I was wondering why this might be the case.