Question

Consider the 2 nd-order partial differential equation for the scalar field u(x,y)

−( ∂x²/∂²u+ ∂y²/∂²u )=
2π² sin(πx)sin(πy) in Ω on a unit square domain Ω=[0,1]×[0,1] and with zero essential boundary conditions on all boundaries
u=0, on ∂Ω=Γ
1. Using the weighted residual method, develop the weak-form for this boundary value problem.

Answer 1

The weighted residual method will be utilized to develop a weak form of the given partial differential equation, incorporating the specified zero boundary conditions and using integration by parts to account for the second derivatives.

Step-by-step explanation:

The question relates to the application of the weighted residual method to develop a weak form of a second-order partial differential equation for a scalar field u(x,y) governed by a specific wave equation in a unit square domain. With the given boundary conditions of u=0 on ∆Ω=Γ1, the weak form essentially involves integrating the product of the equation with a test function over the domain and applying integration by parts, taking into account these boundary conditions.

To derive the weak form, one would multiply the differential equation by a test function v(x,y), which also satisfies the boundary conditions (v=0 on the boundary öΩ), and integrate over the domain Ω. However, the provided references seem to be unrelated to the question since they discuss various unrelated physical phenomena and equations.

The appropriate weak form, when following the weighted residual approach and considering the boundary conditions, would involve functions that satisfy the Dirichlet boundary conditions (u=0 on öΩ) and yield an integral form of the given differential equation after using integration by parts to remove the second derivatives.