2.12.1 Cauchy Riemann PDE with Prescribe the values of \(u\) and \(v\) on the \(x\) axis

problem number 106

From Mathematica DSolve helps pages.

Solve for \(u(x,y),v(x,y\) \begin {align*} \frac {\partial u}{\partial x} &= \frac {\partial v}{\partial y}\\ \frac {\partial u}{\partial y} &= -\frac {\partial v}{\partial x} \end {align*}

With boundary conditions \begin {align*} u(x,0)&=x^3 \\ v(x,0)&=0 \end {align*}

Mathematica ✓

ClearAll["Global`*"]; 
ClearAll[u, v, x, y]; 
 pde1 = D[u[x, y], x] == D[v[x, y], y]; 
 pde2 = D[u[x, y], y] == -D[v[x, y], x]; 
bc  = {u[x, 0] == x^3, v[x, 0] == 0}; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde1, pde2, bc}, {u[x, y], v[x, y]}, {x, y}], 60*10]];
 

\[\left \{\left \{u(x,y)\to x^3-3 x y^2,v(x,y)\to 3 x^2 y-y^3\right \}\right \}\]

Maple ✓

restart; 
pde1:= diff(u(x,y),y)=diff(v(x,y),x); 
pde2:= diff(u(x,y),x)=-diff(v(x,y),y); 
bc  := u(x,0)=x^3,v(x,0)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde1,pde2,bc],[u(x,y),v(x,y)])),output='realtime'));
 

\[\{u \left (x , y\right ) = x^{3}-3 x y^{2}, v \left (x , y\right ) = -3 x^{2} y +y^{3}\}\]

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