Method of Characteristics Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Sophia Torres (Solvelet student)

Solve the first-order partial differential equation using the method of characteristics:

u_t + u_x = 0

with the initial condition

u(x,0) = f(x)

Solution

To solve the first-order partial differential equation

u_t + u_x = 0

with the initial condition

u(x,0) = f(x)

, we use the method of characteristics. 1. Rewrite the PDE in characteristic form:

\frac{dt}{1} = \frac{dx}{1} = \frac{du}{0}

2. Integrate the first pair of equations:

\frac{dt}{1} = \frac{dx}{1} \implies t - x = \text{constant} \implies x - t = c

3. The characteristics are lines

x - t = c

, where

c

is a constant. 4. Along each characteristic,

u

is constant since

\frac{du}{0} = 0

. 5. Express the solution in terms of the initial condition:

u(x,t) = f(x-t)

Therefore, the solution to the PDE is:

u(x,t) = f(x-t)

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Sophia Taylor on Solvelet

1. Solve the first-order partial differential equation

\frac{{\partial u}}{{\partial t}} + u \frac{{\partial u}}{{\partial x}} = 0

using the method of characteristics, where

u(x, 0) = f(x)

.2. Determine the characteristic curves associated with the first-order partial differential equation

\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = 0

DefinitionThe method of characteristics breaks up the partial differential equation into ordinary differential equations along curves called characteristics. This results to be very efficient when treating first-order PDEs. Example: To solve the PDE ut+cux=0, we find characteristics by solving dtdx=c, leading to solutions along lines x−ct=constant.

Related Calculators

One-variable linear equations Calculator FOIL Method Calculator Cramers Rule Calculator Multiply powers of the same base Calculator