hi guys....new image that I am very proud of....utilzing the number theoretic argument λ (n)

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December 10 2017

2
0

hi guys....new image that I am very proud of....utilzing the number theoretic argument λ (n)

December 06 2017

0
11

Let us consider

sol1 := dsolve({diff(y(x), x) = solve((1/2)*(diff(y(x), x))^2 = (1-ln(y(x)^2))*y(x)^2, diff(y(x), x))[1], y(0) = 1}, numeric); sol1 := proc(x_rkf45) ... end proc

The problem under consideration has the symbolic solution:

sol2 := dsolve({diff(y(x), x) = solve((1/2)*(diff(y(x), x))^2 = (1-ln(y(x)^2))*y(x)^2, diff(y(x), x))[1], y(0) = 1}); sol2 := y(x) = exp(x*sqrt(2)-x^2)

Let us compare the plots of sol1 and sol2 (which should coincide):

A := plots:-odeplot(sol1, x = 0 .. 1, color = navy, style = point): B := plot(rhs(sol2), x = 0 .. 1, color = red): plots:-display([A, B]);

The plots* differ* after approximately 0.707. Bug_in_dsolve_numeric.mw

Edit. The title and one of the tags.

December 05 2017

10
7

My co-author and I recently published the 3^{rd} edition of our finite element book^{1} utilizing routines written with MAPLE. In this latest edition, we include a chapter on the meshless method. The meshless method is a unique numerical method for solving PDEs. The finite element method requires the establishment of a mesh associated with node points. Consideration must be given in establishing a good mesh (and minimizing the bandwidth associated with node numbering). The meshless method does not require a mesh to connect nodes. The following excerpt describes the application of the meshless method for a simple 1-D heat transfer simulation using six nodes.

Consider the 1-D expression for heat transfer in a bar defined by the relation^{2}

(1)

where the 1-D domain is bounded by 0 ≤ *x* ≤ *L*. The exact solution to this problem is

(2)

with the exact derivative of the temperature given by

(3)

In order to solve the 1-D problem, a multiquadric (MQ) radial basis function (RBF) is used

(4)

where *r(x, x _{j})* is the radial (Euclidean) distance from the expansion point

(5)

that will be used to solve Eq. (1). Other types of RBFs are available; the MQ is accurate and popular.

A global expansion for the 1-D temperature can be expressed as

(6)

with the second derivative of the temperature given as

(7)

Introducing the RBF expansion for the terms in the governing equation, and collocating at the interior points, we obtain