Cauchy Riemann PDE’s

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

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*}

Mathematica ✓

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 ✓

 
x:='x'; y:='y';u:='u'; 
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'));

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

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33.2 Cauchy Riemann PDE With extra term on right side

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} + y \end {align*}

Mathematica ✗

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] + y; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[{pde1, pde2}, {u[x, y], v[x, y]}, {x, y}], 60*10]];

\[ \text {Failed} \]

Maple ✓

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

\[ \left \{ u \left ( x,y \right ) =-i{\it \_F1} \left ( y-ix \right ) +i{\it \_F2} \left ( y+ix \right ) +yx+{\it \_C1},v \left ( x,y \right ) ={\it \_F1} \left ( y-ix \right ) +{\it \_F2} \left ( y+ix \right ) +1/2\,{x}^{2} \right \} \]

33 Cauchy Riemann PDE’s

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

33.2 Cauchy Riemann PDE With extra term on right side