Method of Lines Calculator

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Example

Created on 2024-06-20Asked by Theodore Ramirez (Solvelet student)

Use the method of lines to solve the heat equation

u_t = k u_{xx}

with the initial condition

u(x,0) = \sin(\pi x)

and boundary conditions

u(0,t) = 0

and

u(1,t) = 0

Solution

To solve the heat equation

u_t = k u_{xx}

with the initial condition

u(x,0) = \sin(\pi x)

and boundary conditions

u(0,t) = 0

and

u(1,t) = 0

, we use the method of lines. 1. Discretize the spatial domain:

x_i = i \Delta x, \quad i = 0, 1, \ldots, N, \quad \Delta x = \frac{1}{N}

2. Approximate the spatial derivative

u_{xx}

using finite differences:

u_{xx} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{(\Delta x)^2}

3. Convert the PDE into a system of ODEs:

\frac{du_i}{dt} = k \frac{u_{i+1} - 2u_i + u_{i-1}}{(\Delta x)^2}, \quad i = 1, 2, \ldots, N-1

4. Apply the initial condition:

u_i(0) = \sin(\pi x_i)

5. Apply the boundary conditions:

u_0(t) = 0, \quad u_N(t) = 0

6. Solve the system of ODEs using a numerical method such as the Euler method or Runge-Kutta method. The solution

u(x,t)

can be approximated by solving the system of ODEs and interpolating the values. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Logan Gonzalez on Solvelet

1. Solve the partial differential equation

\frac{{\partial u}}{{\partial t}} = \frac{{\partial^2 u}}{{\partial x^2}}

using the method of lines with finite difference approximation, where

u(x, 0) = f(x)

and

u(0, t) = u(L, t) = 0

.2. Analyze the stability of the method of lines for solving the advection-diffusion equation

\frac{\partial u}{\partial t} = c \frac{\partial u}{\partial x} + D \frac{\partial^2 u}{\partial x^2}

with

c > 0

and

D > 0

DefinitionThe method of lines is a technique for solving partial differential equations by discretizing the PDE and converting it to a system of ordinary differential equations. These ODEs are solved by conventional methods. The heat equation (in the simplest 1-D case) is ∂tu=k∂2xuxx, and discretizing x allows one to solve this system of ODEs over time.

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