% Script: TestPar61
% Description:  Contains test problem for use by TestParab.m
% Try alpha = 1, beta = 1, d = pi/3 for this example
% This example is used in the paper related to 6.1

% Problem parameters
global A = 1.0; % coefficient of c(0) in BC
global B = -0.5; % coefficient of c'(0) in BC
global edecay = 1; % exponential decay (0=false)

% DEFINE EXACT SOLUTION

function retval = csoln(x,t)
% usage: csoln(x,t)
% description: this computes the solution function csoln(x)

retval = (1-cos(pi*t/4))*exp(-t*x^2/8);

endfunction

% DEFINE THE COEFFICIENTS:

function retval = D(x)
% usage: D(x)
% description: this computes the fcn D(x)
% Reminder: If you change this, you must also change Dp

retval =  1 - exp(-2*x)/2;

endfunction

%

function retval = Dp(x)
% usage: Dp(x)
% description: this computes the fcn D'(x)

retval =  exp(-2*x);

endfunction

%

function retval = v(x)
% usage: v(x)
% description: this computes the fcn v(x)
% Reminder: If you change this, you must also change vp

retval = 1;

endfunction

%

function retval = vp(x)
% usage: vp(x)
% description: this computes the fcn v'(x)

retval = 0;

endfunction

%

function retval = lam(x)
% usage: lam(x)
% description: this computes the fcn lam(x)

retval = 1;

endfunction

%

function retval = f(x,t)
% usage: f(x,t)
% description: this computes the fcn f(x,t)

% computer generated by Maple
      t1 = x*x;
      t2 = t*t1;
      t4 = exp(-t2/8.0);
      t6 = 0.3141592653589793E1*t/4.0;
      t7 = cos(t6);
      t11 = exp(-2.0*x);
      t12 = t11*t;
      t15 = t1*x;
      t18 = t7*x;
      t20 = t1*t7;
      t22 = t*t;
      t23 = t22*t1;
      t31 = t22*t15;
      t34 = t11*t7;
      t38 = -32.0-8.0*t2*t7-4.0*t12*x-4.0*t15*t7+32.0*t18-8.0*t20+ ...
2.0*t23+4.0*t12-t23*t11-8.0*t12*t1-4.0*t12*t7-2.0*t31*t7+t31*t34+4.0*t12*t18;
      t50 = sin(t6);
      t60 = t23*t34+8.0*t12*t20+8.0*t2+8.0*t1+4.0*t15-8.0*t-32.0*x+ ...
2.0*t31+8.0*t*t7-8.0*x*t50*0.3141592653589793E1-2.0*t23*t7- ...
16.0*t50*0.3141592653589793E1-t31*t11+32.0*t7;
retval  = -t4*(t38+t60)/(1.0+x)/32.0;

endfunction

%

function retval = rho(x)
% usage: rho(x)
% description: this computes the fcn rho(x)

retval = (2+x)/(1+x);

endfunction

%

function retval = G(t)
% usage: G(t)
% description: this computes the fcn boundary term G(t)

retval =1 - cos(pi*t/4);

endfunction

%

function retval = g(x)
% usage: g(x)
% description: this computes the initial condition g(x)

retval = csoln(x,0);

endfunction

% DEFINE THE WEIGHT FUNCTION

function retval = p(x)
% REMINDER:  p(0) must equal 1
% usage: p(x)
% description: this computes the weight fcn p(x)

% If you change this, you must also change pp(x)
retval = exp(-x^2);
%retval = exp(-x);

endfunction

%

function retval = pp(x)
% usage: pp(x)
% description: this computes the bdy fcn p'(x)

retval = -2*x*exp(-x^2);
%retval = -exp(-x);

endfunction

%