How can find weak form to d^4u/dx^4+du/dx=f (x)
By using finite element
To find the weak form of the differential equation, d^4u/dx^4 + du/dx = f(x), using the finite element method, we need to follow these steps:
1. Define the problem domain: Make sure you have a clear understanding of the domain over which the differential equation is defined. Identify the boundaries of the domain and any boundary conditions that may be given.
2. Approximate the solution: In the finite element method, we approximate the solution u(x) by using a set of basis functions or shape functions. These functions are typically piecewise-defined over the domain.
3. Multiply the differential equation by a test function: Choose a suitable test function v(x) to multiply the differential equation, so that it becomes a product of u(x) and v(x). The test function should satisfy certain conditions, such as being continuous and having sufficient smoothness.
4. Integrate by parts: Apply integration by parts to the terms involving derivatives in the differential equation. This allows us to shift the derivatives from u(x) to the test function v(x), resulting in a more convenient form of the equation.
5. Apply the principle of weak formulation: The principle of weak formulation states that the differential equation should hold for all test functions in a suitable function space. This leads to a set of equations known as the weak form, which is a variational problem.
6. Apply boundary conditions: Incorporate the boundary conditions into the weak form by modifying the test functions or using additional constraints. This ensures that the solution satisfies the prescribed conditions at the domain boundaries.
7. Discretize the domain: Divide the problem domain into smaller subdomains or elements. Each element is typically represented by a set of nodes and has its own local coordinate system.
8. Define shape functions: Associate shape functions with each element. These functions represent the polynomial approximations within the element and are used to interpolate the values of the unknown function u(x). The shape functions must satisfy the continuity requirements at the element boundaries.
9. Express the weak form in terms of shape functions: Substitute the approximated solution u(x) in terms of the shape functions into the weak form derived in step 5. Perform necessary integrations and manipulations to evaluate the weak form in terms of the shape functions.
10. Apply finite element basis functions: Express the test function v(x) in terms of the same shape functions used to approximate u(x). This enables us to write the weak form as a matrix equation, involving the unknown coefficients associated with the shape functions.
11. Solve the resulting system of equations: By solving the system of equations obtained in the previous step, you can determine the values of the unknown coefficients associated with the shape functions. These coefficients represent the approximation of the solution u(x) within each element.
12. Assemble the solution: Combine the element solutions to obtain the complete solution u(x) over the entire problem domain. This involves applying appropriate boundary conditions, enforcing continuity at element boundaries, and ensuring that the overall solution satisfies the original differential equation.
By following these steps, you can find the weak form of the given differential equation using the finite element method. Keep in mind that the specific implementation may vary depending on the problem and the chosen numerical method.