Problem 1

Plot3D[Sin[x]-Sin[y],{x,-20,20},{y,-20,20},PlotRange->All,ColorFunction->"BlueGreenYellow"]
ContourPlot[Sin[x]-Sin[y],{x,-20,20},{y,-20,20},PlotLegends->Automatic]

Plot3D[(1-x^2) (1-y^2),{x,-50,50},{y,-50,50},PlotRange->All,ColorFunction->"BlueGreenYellow"]
ContourPlot[(1-x^2) (1-y^2),{x,-50,50},{y,-50,50},PlotLegends->Automatic]

Problem 2

ContourPlot[(x^2-y^2)/(x^2+y^2),{x,-20,20},{y,-20,20},PlotLegends->Automatic]

ContourPlot[(x y)/(x^2+y^2),{x,-20,20},{y,-20,20},PlotLegends->Automatic]

Problem 3

ContourPlot[(3x^2 y)/(x^2+y^2),{x,-20,20},{y,-20,20},PlotLegends->Automatic]

Problem 4

Clear[x,y];
Manipulate[
Show[{Plot3D[Log[x-2 y],{x,3-δ,3+δ},{y,1-δ,1+δ}],ListPointPlot3D[{{3,1,0}}],Plot3D[x-2y-1,{x,3-δ,3+δ},{y,1-δ,1+δ},ColorFunction->"Rainbow"]},PlotRange-> {-2,2}]
,{δ,2,0.1,0.1}]

Problem 8

Ω = Disk[{10,10},10];
region=RegionProduct[Ω,Line[{{0},{40}}]];
diff=1;
lhs=D[u[x,y,t],{t,1}]- diff D[u[x,y,t],{x,2}]-diff D[u[x,y,t],{y,2}];
init =u[x,y,0]==Piecewise[{{10,(x-10)^2+(y-10)^2 ≤ 1},{0,True}}];
Clear[u,x,t];
result=NDSolveValue[{lhs==NeumannValue[0,0≤ x≤ 20 && t>0],init},u,{x,y,t}∈ region ];
Manipulate[ Quiet[Show[RegionPlot[Ω], DensityPlot[result[x,y,t],{x,y} ∈Ω,PlotRange->{0,10},PlotLegends->Automatic]]],{t,0,40,0.1}]