Solve using the method of characteristics: u_x+u_y=u^2 in U and u=g on boundary of U

The given partial differential equation is:

\[
u_x + u_y = u^2
\]
with the boundary condition \( u = g \) on the boundary of \( U \).

### Step-by-Step Solution Using the Method of Characteristics:

We aim to solve the first-order nonlinear PDE by employing the **method of characteristics**. The idea is to convert the PDE into a system of ordinary differential equations (ODEs) along certain curves, called characteristic curves, where the PDE simplifies.

1. **Characteristic Equations:**

The PDE can be written as:

\[
\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = u^2
\]

This is a quasilinear equation. The method of characteristics leads to the following system of ODEs:

\[
\frac{dx}{ds} = 1, \quad \frac{dy}{ds} = 1, \quad \frac{du}{ds} = u^2
\]

Here, \( s \) is the parameter along the characteristic curves. These equations describe the propagation of \( u \) along the characteristics.

2. **Solve the First Two Characteristic Equations:**

The first two equations describe the characteristic curves:

\[
\frac{dx}{ds} = 1 \quad \text{and} \quad \frac{dy}{ds} = 1
\]

From these, we get:

\[
x = s + C_1 \quad \text{and} \quad y = s + C_2
\]

Thus, along the characteristics, we have:

\[
x - y = C
\]

where \( C \) is a constant, representing different characteristics.

3. **Solve the Third Characteristic Equation:**

The third characteristic equation is:

\[
\frac{du}{ds} = u^2
\]

This is a separable ODE. We can solve it as follows:

\[
\frac{du}{u^2} = ds
\]

Integrating both sides:

\[
-\frac{1}{u} = s + C_3
\]

Thus, the solution for \( u \) along the characteristics is:

\[
u = \frac{-1}{s + C_3}
\]

4. **Determine the Solution in Terms of \( x \) and \( y \):**

Recall that along the characteristics, \( s = x - C_1 \) and \( s = y - C_2 \), so we can express \( s \) in terms of \( x \) and \( y \). Using the boundary condition \( u = g \) on \( x - y = C \), we can solve for \( C_3 \) in terms of the boundary data.

The final solution will depend on the boundary condition \( u = g \) at \( x - y = C \).

Step-by-step solution for solving the partial differential equation u_x + u_y = u^2 using the method of characteristics. This solution includes solving a system of ordinary differential equations and applying boundary conditions. Suitable for undergraduate-level students in math or engineering.

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