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11.4 Sample ‘apde’ ¶

Comparison of advanced PDE solver (apde) and ordinary one (pde).

MGL code:

ranges -1 1 0 2 0 2
new ar 256 'exp(-2*(x+0.0)^2)'
new ai 256

apde res1 'exp(-x^2-p^2)' ar ai 0.01:transpose res1
pde res2 'exp(-x^2-p^2)' ar ai 0.01

subplot 1 2 0 '_':title 'Advanced PDE solver'
ranges 0 2 -1 1:crange res1
dens res1:box
axis:xlabel '\i z':ylabel '\i x'
text -0.5 0.2 'i\partial_z\i u = exp(-\i x^2+\partial_x^2)[\i u]' 'y'

subplot 1 2 1 '_':title 'Simplified PDE solver'
dens res2:box
axis:xlabel '\i z':ylabel '\i x'
text -0.5 0.2 'i\partial_z\i u \approx\ exp(-\i x^2)\i u+exp(\partial_x^2)[\i u]' 'y'

C++ code:

void smgl_apde(mglGraph *gr)
{
	gr->SetRanges(-1,1,0,2,0,2);
	mglData ar(256), ai(256);	gr->Fill(ar,"exp(-2*(x+0.0)^2)");

	mglData res1(gr->APDE("exp(-x^2-p^2)",ar,ai,0.01));	res1.Transpose();
	mglData res2(gr->PDE("exp(-x^2-p^2)",ar,ai,0.01));

	gr->SubPlot(1,2,0,"_");	gr->Title("Advanced PDE solver");
	gr->SetRanges(0,2,-1,1);	gr->SetRange('c',res1);
	gr->Dens(res1);	gr->Axis();	gr->Box();
	gr->Label('x',"\\i z");	gr->Label('y',"\\i x");
	gr->Puts(mglPoint(-0.5,0.2),"i\\partial_z\\i u = exp(-\\i x^2+\\partial_x^2)[\\i u]","y");

	gr->SubPlot(1,2,1,"_");	gr->Title("Simplified PDE solver");
	gr->Dens(res2);	gr->Axis();	gr->Box();
	gr->Label('x',"\\i z");	gr->Label('y',"\\i x");
	gr->Puts(mglPoint(-0.5,0.2),"i\\partial_z\\i u \\approx\\ exp(-\\i x^2)\\i u+exp(\\partial_x^2)[\\i u]","y");
}