/* Last updated: December 19, 2020 by Hereman in Boulder */
/* For use with symmgrp2020.max under Maxima */

/* File name: Fokker-Planck-Kolmogorov-run3-exp-rund-Dec19-2020-wh.dat. */
/* Data file for the Fokker-Planck-Kolmogorov equation with varying coefficient given in the online manuscript Faruk Gungor. */

/* CASE: pp(x) = exp(x) */
/* The info below is based on the determining equations obtained in Run 2 */

/* u_t - u_xx + pp(x) u_y = 0 */
/* Assign u[1] to u, x[1] to x, x[2] to y, and x[3] to t, pp is a function of x only. */

/* number of independent variables: */
p : 3$

/* number of dependent variables: */
q : 1$

/* number of equations in the system: */
m : 1$

warnings: false$
parameters : []$
sublisteqs : [all]$
subst_deriv_of_vi: true$
highest_derivatives : all$

/* information is give in this run. */
info_given : true$

/* careful the symbol p is reserved for the number of independent variables. So, use pp. */

depends(pp,x[1]);

pp : exp(x[1]);

depends([eta1,f1,f2],[x[1],x[2],x[3]]);
depends([eta2,eta3],[x[2],x[3]]);

depends(hh1,x[2]); 
depends(ii1,[x[2],x[3]]); 

/* f2 satisfies the Fokker-Planck-Kolmogorov equation itself -- determining an infinite symmetry */
gradef(f2,x[3],-pp*diff(f2,x[2])+diff(f2,x[1],2));

/* Take it out again to see the constraint on f1 explicitly. */
/* gradef(f1,x[3],-pp*diff(f1,x[2])+diff(f1,x[1],2)); */

/* For the generic (or principal) case: Taking the determining equations from Run 1, we split by powers of pp */
/* and set the coefficients of derivatives also equal to zero. */
/* We solved this by hand. We just test the result here. */
/* For the generic case, i.e., valid for any function pp(x) : */
/* eta1 : 0; */
/* eta2 : c2; */
/* eta3 : c1; */
/* f1 : c3; */

hh1 : ccc2*x[2]+ccc3;
ii1 : ccc4;

eta3 : ccc1;
eta2 : hh1 + ccc5*x[2]^2;
f1 : -(1/2)*x[1]*diff(hh1,x[2],2)+ii1;

/* This is based on the results of Run 2. We are using it this time: */
eta1 : -(pp^2/diff(pp,x[1]))*diff(eta3,x[2])-(pp/diff(pp,x[1]))*(diff(eta3,x[3])-diff(eta2,x[2]))+(1/diff(pp,x[1]))*diff(eta2,x[3])+2*ccc5*x[2];

phi1 : ( f1 -ccc5*(exp(x[1])+x[2]) )*u[1]+f2;

e1 : u[1,[0,0,1]] - u[1,[2,0,0]] + pp*u[1,[0,1,0]];
/* pp is function of x only */

v1 : u[1,[0,0,1]]; 

/* Running the code with either v1 = u[1,[0,0,1]] or v1 = u[1,[2,0,0] results in determining equations which are not completely split! */
/* Needs TO BE INVESTIGATED */
/* v1 : u[1,[2,0,0]]; */
/* Current fix: Give gradef for f1 based on information from run1. */

/* end of data file Fokker-Planck-Kolmogorov-run3-exp-rund-Dec19-2020-wh.dat. */