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

     

     

    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.

    My co-author and I recently published the 3rd edition of our finite element book1 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 relation2

         (1)

    where the 1-D domain is bounded by 0 ≤ xL. 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, xj) is the radial (Euclidean) distance from the expansion point (xj) to any point (x) , c is a shape parameter that controls the flatness of the RBF and is set by the user, and n is an integer. With n = 1, we retrieve the inverse multiquadric

         (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