Please I need a solution with detailed steps of the following 1D heat equation using RBF-FD. when T1=0.1 and T2=0.2
1D Heat Equation:
∂u/∂t = 0.5* ∂²u/∂x², for 0 ≤ x ≤ 1 and 0 ≤ t ≤ T.
Parameters:
Δt = 0.1 (Time step size)
Δx = 0.25 (Spatial step size)
Initial Condition: u(x, 0) = sin(π * x)
Boundary Conditions: u(0, t) = u(1, t) = 0
To solve the 1D heat equation using RBF-FD (Radial Basis Function Finite Differences), we will discretize the equation in space and time using finite differences and then use a numerical solver to approximate the solution. Here are the detailed steps to solve the problem:
Step 1: Discretization
- Divide the spatial domain [0, 1] into grid points with a spatial step size Δx = 0.25. Let xi be the spatial grid points: xi = i * Δx, where i = 0, 1, 2, ..., N (N is the number of spatial grid points).
- Divide the time domain [0, T] into time steps with a time step size Δt = 0.1. Let tk be the time steps: tk = k * Δt, where k = 0, 1, 2, ..., M (M is the number of time steps).
- Define the initial condition: u(xi, 0) = sin(π * xi).
Step 2: RBF-FD Approximation
- For each spatial grid point xi, calculate the weights for its neighboring points using a radial basis function (RBF).
- For example, you can use the Gaussian RBF: ϕ(r) = exp(-ε * r^2), where ε is a user-defined parameter, and r is the distance between xi and its neighboring points.
Step 3: Time Integration
- Use an explicit time-stepping method to solve the time-dependent part of the equation.
- For each time step tk, update the solution at each spatial grid point xi using the finite difference approximation for the time derivative: ∂u(xi, tk)/∂t ≈ [u(xi, tk) - u(xi, tk-1)] / Δt.
- Combine the RBF-FD approximation for the spatial derivatives with the time-stepping method to solve for the solution at each time step.
Step 4: Boundary Conditions
- Apply the boundary conditions u(0, t) = u(1, t) = 0 at each time step by setting the values of u(0, tk) and u(1, tk) to 0.
Step 5: Iterate and Solve
- Start with the initial condition u(xi, 0) = sin(π * xi) at t = 0.
- Use the RBF-FD approximation and time-stepping method to calculate the solution at each subsequent time step until you reach the final time T.
Step 6: Post-Processing
- Once you have calculated the solution at all time steps, you can analyze and visualize the results as needed.
Please note that implementing RBF-FD can be more involved than traditional finite difference methods, and the accuracy and stability of the method can depend on the choice of RBF, its parameters, and the spatial and temporal resolutions. Additionally, as the number of spatial grid points and time steps increases, the computation time can also increase significantly.
Keep in mind that this is a general outline of the approach. Implementing the RBF-FD method requires a good understanding of numerical methods and programming skills. You may refer to literature or research papers on RBF-FD for more specific details and examples.
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