Fri May 9 11:10:39 MDT 1997 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Partial Differential Equations ---------- *) 0. Second In[6]:= (* A very simple PDE => g(x) + h(y) for arbitrary functions g and h *) 0. Second In[7]:= de = D[f[x, y], x, y] == 0 0. Second (1,1) Out[7]= f [x, y] == 0 In[8]:= DSolve[de, f[x, y], {x, y}] 0.15 Second Out[8]= {{f[x, y] -> -(DSolve`DSolveDump`b$8 x y) + C[1][x] + C[2][y]}} In[9]:= (* Heat equation: the fundamental solution is 1/sqrt(4 pi t)\ > exp(-x^2/[4 t]). If f(x, t) and a(x, t) are solutions, the most general solution obtainable from f(x, t) by group transformations is of the form u(x, t) = a(x, t) + 1/sqrt(1 + 4 e6 t) exp(e3 - [e5 x + e6 x^2 - e5^2 t]/[1 + 4 e6 t]) f([e^(-e4) (x - 2 e5 t)]/[1 + 4 e6 t] - e1, [e^(-2 e4) t]/[1 + 4 e6 t] -\ > e2) See Peter J. Olver, _Applications of Lie Groups to Differential\ > Equations_, Second Edition, Springer Verlag, 1993, p. 120 (an excellent book). See\ > also Heat.math *) 0. Second In[10]:= de = D[u[x, t], t] == D[u[x, t], {x, 2}] 0.01 Second (0,1) (2,0) Out[10]= u [x, t] == u [x, t] In[11]:= DSolve[de, u[x, t], {x, t}] DSolve::pde: Partial differential equation may not have a general solution. Try loading Calculus`DSolveIntegrals` to find special solutions. 0.04 Second (0,1) (2,0) Out[11]= DSolve[u [x, t] == u [x, t], u[x, t], {x, t}] In[12]:= << Calculus`DSolveIntegrals` 0.26 Second In[13]:= CompleteIntegral[de, u[x, t], {x, t}] 0.01 Second (0,1) (2,0) Out[13]= CompleteIntegral[u [x, t] == u [x, t], u[x, t], {x, t}, > IntegralConstants -> B] In[14]:= (* Potential equation on a circular disk---a separable PDE => v(r, theta) = a[0] + sum(a[n] r^n cos(n theta), n = 1..infinity) + sum(b[n] r^n sin(n theta), n = 1..infinity) *) 0. Second In[15]:= de = (1/r * D[r * D[v[r, theta], r], r] + 1/r^2 * D[v[r, theta], {theta, 2}] == 0) 0.01 Second (0,2) (1,0) (2,0) v [r, theta] v [r, theta] + r v [r, theta] Out[15]= ---------------- + ------------------------------------- == 0 2 r r In[16]:= DSolve[de, v[r, theta], {r, theta}] 0.05 Second (0,2) (1,0) (2,0) v [r, theta] v [r, theta] + r v [r, theta] Out[16]= DSolve[---------------- + ------------------------------------- == 2 r r > 0, v[r, theta], {r, theta}] In[17]:= CompleteIntegral[de, v[r, theta], {r, theta}] 0.01 Second (0,2) v [r, theta] Out[17]= CompleteIntegral[---------------- + 2 r (1,0) (2,0) v [r, theta] + r v [r, theta] > ------------------------------------- == 0, v[r, theta], {r, theta}, r > IntegralConstants -> B] In[18]:= Clear[de] 0. Second In[19]:= (* ---------- Quit ---------- *) 0. Second In[20]:= Quit[] real 6.13 user 1.33 sys 0.36