# Tag Info

14

A first step would be to implement a convenience function that can automatically apply the method of separation of variables to separable types of equations. To show that the steps could in principle be automated, let me repeat basically the same calculation that I did for cylindrical coordinates with only slight modifications to the heat equation: ...

10

Here is extensions to @Jens answer (I think) also relying on possible separation of variable. I is not meant as an independent answer, but complements it. First extend his answer to 2D ClearAll[pt, px, x, t, p]; operator = Function[p, D[p, t] - Δ D[p, x, x] - Δ D[p, y, y]]; ansatz = pt[t] px[x] py[y]; pde2 = Expand[Apply[Subtract, ...

5

Good question; the notion of a tensorial (covariant) derivative is something that is missing in Mathematica AFAIK. I can think of two ways to proceed: Option 1 One way is to overload the TensorRank, TensorDimensions, and TensorSymmetry functions for patterns that have head CD: CD /: TensorRank[CD[tensor_]] := TensorRank[tensor] + 1 CD /: ...

5

The differential operator in the first form can be written as dd1[n_] := (Sum[a[k] D[#, {t, k}], {k, 0, n}]) & and is applied for example as dd1[1][x[t]] a[0] x[t] + a[1] Derivative[1][x][t] and dd1[2] @x[t] a[0] x[t] + a[1] Derivative[1][x][t] + a[2] (x^\[Prime]\[Prime])[t] In the second (product) form we would perhaps try d2[n_] := a[n] ...

4

The Package HypExp does exactly that. Here is the link to paper for what I believe was the last extension. After digging around a bit, the package files should be available here ( Edit freely available link) Several years ago, there has been some work on the simplification of polylogarithms into a Hopt Algebras, which simplifies the reduction of the ...

4

I assume you are looking for a pretty specific answer. If this is less information than you are asking for feel free to comment or edit the question and I will expand. As you know, the frontend represents expressions using boxes. These are wrappers that are concerned with appearance, and are peripheral to core symbolic evaluation. Therefore, each step of ...

4

Only a suggestion, not a full answer: As far as I can see the ISC returns Maple code. Therefore, after importing the output of the website as string into Mathematica, the biggest challenge is to convert the output from Maple to Mathematica code. A quick google search reveals that there is a package in the MathLibrary. I guess chances are good that the ...

3

Here is another way that uses the Graphics object directly: gr = ParametricPlot3D[{Cos[u], Sin[u] + Cos[v], Sin[v]}, {u, 0, 2 Pi}, {v, -Pi, Pi}] We discretize the graphics using DiscretizeGraphics mr = DiscretizeGraphics[Normal[gr /. (Lighting -> _) :> Lighting -> Automatic]] We compute the convex hull hull = ...

2

This is also not an answer but a brief study in \$Version "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" which might be of interest. It considers three methods of calculating the requested probability. It shows that in this version there is no negative probability but there is still a "critical number" which amounts to 8. For n = 8 the ...

2

You can set the multiplication symbol in Preferences->Appearance->Numbers->Multiplication

2

Your integral is very unlikely to exist in terms of elementary functions. In particular, it involves terms of the form $$\int\exp\left[-\frac12\sqrt{a\, \text{poly}(\xi)+b \xi^{0.998906}}\right]\text d\xi,$$ which is very unfriendly as regards symbolic integration. Note that in general symbolic integration is not possible; do you have some specific reason ...

2

In Version 10, once the points have been obtained as per user21's approach, we can tetrahedralize them directly using DelaunayMesh pf = {Cos[u], Sin[u] + Cos[v], Sin[v]}; pp = ParametricPlot3D[pf, {u, 0, 2 Pi}, {v, -Pi, Pi}] data = Reap[ParametricPlot3D[Sow[pf], {u, 0, 2 Pi}, {v, -Pi, Pi}]][[2, 1]]; pts = Cases[data, {_?NumericQ, _?NumericQ, ...

2

I get a better fit with an asymptotic complexity between n^4 and n^5. I think Det is doing a lot of simplification of symbolic expressions, which may account for some of the increased complexity. nMax = 35; entry[] := RandomInteger[{-9, 9}] + RandomInteger[{-9, 9}]*t findTime[n_] := Block[{m, time, det}, m = Table[entry[], {i, 1, n}, {j, 1, n}]; {time, ...

2

rules = {a b -> p, a c / d -> q} Expand[a (b + 42 c/d)] /. rules (*p + 42 q*)

2

Here is an example of how you can fetch the results, but there is the question of what to do if multiple results are returned. I'm using URLBuild but you could do the same manually if you don't have Mathematica 10. num = 4.17 Cases[ Import[ URLBuild[ "http://isc.carma.newcastle.edu.au/advancedCalc", {"input" -> num}], "XMLObject"], ...

1

Let me start addressing the Green function part of the question. Lets define a Heat equation and its generic solution (see above) operator[p_] := D[p, t] - D[Δ D[p, x], x]; sol = Heat[Δ, -Infinity] and build a general solution via superposition as: sol1 = Integrate[(sol /. x -> x - y) g[y], {y, -Infinity, Infinity}] Plot[sol1 /. g -> ...

1

In a few seconds Mathematica 7.0.1 under Windows returns: (Mass (r1 Rotation + (r1^2 + Rotation^2) ArcTan[Rotation/r1]))/(2 r1^2 Rotation) On the other hand Mathematica 10.0.0 is still working after several minutes. Assuming the above output is correct this seems like a regression.

1

Question How do I prevent Part[] from trying to decompose symbolic expressions when it is evaluated? Mathematica 10 implements something like your listPart (with additional functionality): Indexed: Indexed can be used to indicate components of symbolic vectors, matrices, tensors, etc. When expr is a list, Indexed[expr,i] gives ...

1

I suspect there may not be a closed form expression (I have not looked at it hard enough). If the aim is to not analytical but numerical, I post the following for illustration (apologies if not the intent of question): f[x_, y_] := NIntegrate[ Log2[t + 1] Exp[-(x - Log[t])^2/(2 y^2)]/(Sqrt[2 Pi] t y), {t, 0, Infinity}] rv[a_, b_, n_] := ...

1

@Daniel Lichtblau's comment seems like an answer that is worth putting in an answer: (1) Integrate will not catch conditions that are discrete. (2) As was pointed out already in comments, the result is correct anyway; the singularity is removable (e.g. via Limit). Edit: I might add that GenerateConditions might yield a ConditionalExpression but not a ...

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