(A) Consider the wave equation with c=1, l=1, u(0,t)=0, and u(l,t)=0. The initial data are: f(x)=x(1-x)2, g(x)=sin2(pi x). Find the value of the solution at x=0, t=10, and at x=1/3, t=0. Find the value of the solution at x=1/2, t=2.
(B) Suppose that l=2, c=1/2. Draw the domain of influence for the point (1/2,0), for values of t between 0 and 6. Find the equations for characteristics for t<6. Find the domain of dependence for point (1,3) (graphically and by writing down equations).
(A) To solve the wave equation, we need to find the solution u(x,t) that satisfies the given initial conditions and boundary conditions. Let's go step by step to find the values of the solution at the specified points.
Step 1: Find the general solution for u(x,t)
The general solution for the wave equation is given by u(x,t) = F(x-ct) + G(x+ct), where F and G are arbitrary functions. In this case, we have c=1, l=1, u(0,t)=0, and u(l,t)=0. Therefore, the solution takes the form u(x,t) = F(x-t) + G(x+t).
Step 2: Apply the initial data to find F and G
We are given the initial data f(x) = x(1-x)^2 and g(x) = sin^2(πx). We need to find F and G that satisfy the conditions u(x,0) = f(x) and ∂u/∂t(x,0) = g(x). Plugging in t=0 in the general solution, we get u(x,0) = F(x) + G(x). This should be equal to f(x), so we can equate the two expressions and solve for F(x) + G(x) = f(x).
Equation 1: F(x) + G(x) = f(x)
Taking the derivative of the general solution with respect to t and plugging in t=0, we get ∂u/∂t(x,0) = -F'(x) + G'(x). This should be equal to g(x), so we can equate the two expressions and solve for -F'(x) + G'(x) = g(x).
Equation 2: -F'(x) + G'(x) = g(x)
Solving Equations 1 and 2 will give us F(x) and G(x).
Step 3: Evaluate the solution at the specified points
(a) x=0, t=10
Using the solution u(x,t) = F(x-t) + G(x+t), we can plug in the values x=0 and t=10 to obtain u(0,10) = F(-10) + G(10).
(b) x=1/3, t=0
Similarly, plugging in x=1/3 and t=0 into the solution, we get u(1/3,0) = F(1/3) + G(1/3).
(c) x=1/2, t=2
Again, plugging in x=1/2 and t=2 into the solution, we get u(1/2,2) = F(-3/2) + G(5/2).
To find the values of u(0,10), u(1/3,0), and u(1/2,2), we need to solve Equations 1 and 2 to find F(x) and G(x) using the given initial data.
(B) To draw the domain of influence for the point (1/2,0) for values of t between 0 and 6, follow these steps:
Step 1: Determine the characteristics
The characteristics of the wave equation are defined by dx/dt = c, where c is the wave speed. In this case, we have c = 1/2, so dx/dt = 1/2.
Step 2: Find the equations for the characteristics for t < 6
Plugging c = 1/2 into dx/dt = c, we get dx/dt = 1/2. Integrating both sides with respect to t, we get x = (1/2)t + C, where C is an integration constant. This equation represents the equation for the characteristics.
Step 3: Draw the domain of influence
To draw the domain of influence for the point (1/2,0), we need to consider the characteristics that pass through this point. Setting x = 1/2 and t = 0 in the equation for the characteristics, we can solve for C. This gives us C = 1/2. Therefore, the equation for the characteristic passing through the point (1/2,0) is x = (1/2)t + 1/2.
For values of t between 0 and 6, we can plot the characteristics by choosing different values of t and substituting them into the equation x = (1/2)t + 1/2. This will show the domain of influence for the point (1/2,0) at different time levels.
To find the domain of dependence for the point (1,3), we need to reverse the characteristic equations. The equation x = (1/2)t + C becomes t = 2x - 2C. Since we are looking for t values for a fixed x value, we can consider the equation t = 2x - 2C as the equation for the domain of dependence. Substituting x = 1 and t = 3 into the equation, we can solve for the appropriate value of C. Once we find C, we can rewrite the equation to represent the domain of dependence for the point (1,3).
Drawing the domain of dependence graphically involves plotting the equation t = 2x - 2C for different values of C to determine the range of t values for a given point (x, t).
Note: The solutions to the wave equation and the drawing of the domains depend on the specific values and conditions given in the problem. Please ensure that all input values are considered accurately in order to obtain precise results.