u_t + (u^2)*u_x = 0 with u(x, 0) = sqrt(x)
To solve the given PDE, we will use the method of characteristics.
Let's first find the characteristic equations for this PDE:
dx/dt = 1
du/dt = u^2 * du/dx
From the first equation, we can see that dx = dt. Integrating both sides, we have:
x = t + C1
where C1 is the constant of integration.
Substituting this into the second equation, we have:
du/dt = u^2 * du/dx
du/dt = u^2 * 1/(dx/dt)
du/dt = u^2 * dt/dx
du/u^2 = dt
Integrating both sides, we obtain:
-1/u = t + C2
where C2 is the constant of integration. Rearranging this equation, we have:
u = -1/(t + C2)
Now, we need to find the values of C1 and C2 by considering the initial condition u(x, 0) = sqrt(x). Substituting x = t + C1 and u = sqrt(x) into the equation u = -1/(t + C2), we have:
sqrt(x) = -1/(t + C2)
Squaring both sides, we obtain:
x = 1/(t + C2)^2
Using x = t + C1, we can rewrite this equation as:
t + C1 = 1/(t + C2)^2
Since this equation holds for all values of t, we can equate the corresponding terms:
C1 = 1/(C2^2)
Now, we can substitute this value of C1 into the equation x = t + C1:
x = t + 1/(C2^2)
Finally, let's solve for t in terms of x:
t = x - 1/(C2^2)
Substituting this into the equation u = -1/(t + C2), we have:
u = -1/((x - 1/(C2^2)) + C2)
u = -1/(x + C2 - 1/(C2^2))
Simplifying this expression further is difficult due to the presence of the constant C2. However, we have obtained the general solution to the PDE u_t + (u^2)*u_x = 0.
To solve the given partial differential equation (PDE) u_t + (u^2)*u_x = 0 with the initial condition u(x, 0) = sqrt(x), we can use the method of characteristics. Let's proceed step by step.
Step 1: Determine the characteristic equations:
The characteristic equations are given by:
dx/dt = 1
du/dt = 0
du/dx = u^2
Step 2: Solve the characteristic equations:
From the first equation, we have dx/dt = 1, which implies dx = dt.
From the second equation, we have du/dt = 0, which implies du = 0.
From the third equation, we have du/dx = u^2. Integrating both sides with respect to x, we get:
∫1/du = ∫u^2 dx
=> u^(-1) = x + C
=> u = 1/(x + C), where C is an arbitrary constant.
Step 3: Determine the values of u(x, t):
Using the initial condition u(x, 0) = sqrt(x), we can substitute t = 0 and u = sqrt(x) into the solution obtained in step 2:
u = 1/(x + C) = sqrt(x)
Squaring both sides, we get:
1/(x + C) = x
=> x + C = 1/x
=> x^2 + Cx - 1 = 0
Step 4: Solve for C:
Solving the quadratic equation x^2 + Cx - 1 = 0, we can find the value of C as follows:
Using the quadratic formula,
C = (-b ± √(b^2 - 4ac))/(2a)
=> C = (0 ± √(0^2 - 4(1)(-1)))/(2(1))
=> C = ± √(4/4)
=> C = ± 1
Step 5: Write the general solution:
We found that C = ±1, so the two characteristic curves (or characteristics) are given by u = 1/(x + 1) and u = 1/(x - 1).
The general solution of the PDE can be written as a combination of these two characteristics:
u(x, t) = f(1/(x + 1)) + g(1/(x - 1)), where f and g are arbitrary functions.
Note that we can simplify the general solution further by substituting v = x + 1 and w = x - 1:
u(x, t) = f(1/v) + g(1/w), where v = x + 1 and w = x - 1.
Step 6: Conclusion:
The solution to the given PDE u_t + (u^2)*u_x = 0, subject to the initial condition u(x, 0) = sqrt(x), is u(x, t) = f(1/(x + 1)) + g(1/(x - 1)), where f and g are arbitrary functions.