Exercises

Find the separated solution to each of the following partial differential equations. Assume that k, a, c, and t are constant.

11.  ¶u ¶t =  ¶u ¶x 12.  ¶u ¶t = -k  ¶u ¶x 13.  ¶u ¶x +  ¶u ¶y = 0 14.  ¶u ¶x = -2x  ¶u ¶y 15. Fx+2xFy = 0 16. Fx+3x2Fy = 0 17. ux+ut = u 18.  ¶u ¶x  ¶u ¶y = u 19.  ¶2u ¶x2 +  ¶2u ¶y2 = 0 20.  ¶2V ¶x2 -t  ¶V ¶t -V = 0 21. ut = uxx+u 22. ut = uxx-u 23.  ¶2u ¶t2 -  ¶u ¶x  ¶u ¶t = 0 24.  ¶2u ¶t2 +  ¶u ¶x  ¶u ¶t = 0

25. Show that

u( x,t) =     1 t e -x2/(4t)

is a solution to the heat equation u_t = u_xx.

26. Show that f( x,y,z) = (x²+y²+z²) ^1/2 is a solution to the 3 dimensional Laplace equation

¶2u ¶x2 +  ¶2u ¶y2 +  ¶2u ¶z2 = 0

27. Let i² = -1 and suppose that u( x,y) and v( x,y) are such that

( x+iy)2  =  u( x,y) +i v( x,y)

Find u and v and show that both satisfy Laplace's equation-that is, that

¶2u ¶x2 +  ¶2u ¶y2 = 0    and     ¶2v ¶x2 +  ¶2v ¶y2 = 0

In addition, show that u and v satisfy the Cauchy-Riemann Equations

ux = vy,        uy = -vx

28. Let i² = -1 and suppose that u( x,y) and v( x,y) are such that

( x+iy)4  =  u( x,y) +i v( x,y)

Find u and v and show that both satisfy Laplace's equation-that is, that

¶2u ¶x2 +  ¶2u ¶y2 = 0    and     ¶2v ¶x2 +  ¶2v ¶y2 = 0

In addition, show that u and v satisfy the Cauchy-Riemann Equations

ux = vy,        uy = -vx

29. Suppose that a large population of microorganisms (e.g., bacteria or plankton) is distributed along the x-axis. If u(x,t) is the population per unit length at location x and at time t, then u satisfies a diffusion equation of the form

¶u ¶t = m  ¶2u ¶x2 +ru

where m is the rate of dispersal and r is the birth rate of the microorganisms. If m and r are positive constants, then what is a separated solution of this diffusion equation. (Adapted from Mathematical Models in Biology, Leah Edelstein-Keshet, Random House, 1988, p. 441).

30. Suppose that t denotes time and x denotes the age of a cell in a given population of cells, and let

31. Find the separated solution of the telegraph equation with zero self inductance:

¶2u ¶x2 = RC  ¶u ¶t +RSu

Here u( x,t) is the electrostatic potential at time t at a point x units from one end of a transmission line, and R, C, and S are positive constants representing the resistance, capacitance, and leakage conductance per unit length, respectively.

32. If V( x,t) is the membrane voltage at time t in seconds and at a distance x from the distal (i.e., initial) end of an uniform, cylindrical, unbranched section of a dendrite, then V( x,t) satisfies

d 4Ri  ¶2V ¶2x = Cm  ¶V ¶t +  1 Rm V

(11)

where d is the diameter of the cylindrical dendritic section, R_i is the resistivity of the intracellular fluid, C_m is the membrane capacitance, and R_m is the membrane resistivity.. Find a separated solution to (11) given that C_m, R_m, and R_i are positive constants.

33. In Quantum mechanics, a particle moving in a straight line is said to be in a state y( x,t) if

ó õ b a  | y( x,t) | 2dx

represents the probability of the particle being in the interval [a,b] on the line at time t. If a subatomic particle is traveling in a straight line close to the speed of light, then it's state satisfies the one dimensional Klein-Gordon Equation

¶2y ¶t2  -   ¶2y ¶x2 = ly

where l > 0 is constant. Find the separated solution of the one dimensional Klein-Gordon equation.

34. If a subatomic particle is traveling in a straight line much slower than the speed of light and no forces are acting on that particle, then it's state (as explained in problem 33) satisfies the one dimensional Schrödinger equation of a single free particle.

¶y ¶t = -i  ¶2y ¶x2

(12)

where i² = -1. Find the separated solution of (12) (Hint: you will need to use Euler's identity

eit = cos( t) +isin( t)

35. Show that if u( x,t) and v(x,t) are both solutions to the one dimensional wave equation

¶2u ¶t2 = a2  ¶2u ¶x2

then so also is the function w( x,t) = Au( x,t)+Bv( x,t) where A and B are constants. What does this say about the wave equation?

36. Show that if u( x,y) and v(x,y) are both solutions to Laplace's equation

¶2u ¶x2 +  ¶2u ¶y2 = 0

then so also is the function w( x,y) = Au( x,y)+Bv( x,ty) where A and B are constants. What does this tell us about Laplace's equation?       

37. Suppose that the initial conditions for the guitar string in example 6 are

u( x,0) = sin æ è  x 2 ö ø     and     ¶u ¶t ( x,0) = 0

What are the coefficients b_n in the solution (9) for these initial conditions?

38. Solve the vibrating string problem for the boundary conditions

¶u ¶x ( 0,t) = 0        and         ¶u ¶x ( l,t) = 0

and for the initial conditions u( x,0) = f( x) and u_t( x,0) = 0.

39. Heat Equation I: Find the general solution to the heat equation

¶u ¶t = k  ¶2u ¶x2

subject to the boundary conditions

u( 0,t) = 0        u( p,t) = 0

and to the initial condition u( x,0) = f( x) .

40. Heat Equation II: If the initial condition is u( x,0) = px-x², then what are the Fourier coefficients in the general solution found in exercise 39?

41. Laplace's Equation I: Find the general solution to the Laplace equation

¶2u ¶x2 +  ¶2u ¶y2 = 0

subject to the boundary conditions

u( 0,y) = 0        u( p,y) = 0

and to the initial conditions u( x,0) = sin( x/2) and u_y( x,0) = 0.

42. Laplace's Equation II: If the initial condition is u( x,0) = sin( x/2) , then what are the Fourier coefficients in the general solution found in exercise 41?

43. Write to Learn: In a short essay, explain in your own words why an equation of the form

f( x) = g( t)

implies that both f( x) and g( t) are constant. (x and t are both independent variables).

44. *What is a separated solution of the 2-dimensional wave equation

¶2u ¶t2 = a  ¶2u ¶x2 +b  ¶2u ¶y2

45. *Find the separated solution of the following nonlinear wave equation:

¶u ¶t = cu  ¶u ¶x