Numerical Methods for PDEs Calculator

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

Example

Created on 2024-06-20Asked by Mason Campbell (Solvelet student)

Solve the heat equation

u_t = \alpha u_{xx}

using the finite difference method with 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 = \alpha u_{xx}

using the finite difference method, follow these steps: 1. Discretize the spatial domain with

\Delta x = \frac{1}{N}

and the time domain with

\Delta t

. 2. Use the finite difference approximations:

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

u_t \approx \frac{u_i^{n+1} - u_i^n}{\Delta t}.

3. Substitute into the heat equation:

\frac{u_i^{n+1} - u_i^n}{\Delta t} = \alpha \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\Delta x)^2}.

4. Rearrange to solve for

u_i^{n+1}

u_i^{n+1} = u_i^n + \lambda (u_{i+1}^n - 2u_i^n + u_{i-1}^n),

where

\lambda = \frac{\alpha \Delta t}{(\Delta x)^2}

. 5. Apply initial and boundary conditions:

u_i^0 = \sin(\pi x_i), \quad u_0^n = 0, \quad u_N^n = 0.

6. Iterate over time steps to compute the solution. Therefore, using the finite difference method, the solution to the heat equation can be approximated iteratively using the specified initial and boundary conditions. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Ella Harris on Solvelet

1. Solve the heat equation

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

using the finite difference method with a given initial condition and boundary conditions.2. Approximate the solution to the wave equation

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

using the finite element method with a given initial condition and boundary conditions.

DefinitionThe majority of PDE methods on a numerical computer is based also only on Partial Differential Equations (PDEs) with differences discretizing the equations and approximation of the resulting equations using methods of Finite Difference, Finite Element, and Finite Volume. Finite Difference: u = ukk−1…k0Single equation — decisions at all time levelsThe finite difference method is used to approximate the solution of a parabolic equation such as the heat equation.

Related Calculators

Normal Distribution Calculator Homogeneous and Heterogeneous Calculator Fourier Series Calculator Weight and Mass Calculator

Need topic explanation ? Get video explanation