Nonlinear differential equation problems¶
Contents:
- Nonlinear differential equation problems
- Basic examples using the logistic equation
- Systems of nonlinear algebraic equations
- Linearization at the differential equation level
- Discretization of nonlinear differential equations
- Finite difference discretizations
- Finite element discretizations
- The group finite element method
- Numerical integration of nonlinear terms
- Finite element discretization of a variable coefficient Laplace term
- Picard iteration defined from the variational form
- Newton’s method defined from the variational form
- Multi-dimensional PDE problems
- Exercises
- Problem 1: Determine if equations are nonlinear or not
- Problem 2: Linearize a nonlinear vibration ODE
- Exercise 3: Find the sparsity of the Jacobian
- Exercise 4: Newton’s method for linear problems
- Exercise 5: Differentiate a highly nonlinear term
- Problem 6: Discretize a 1D problem with a nonlinear coefficient
- Problem 7: Linearize a 1D problem with a nonlinear coefficient
- Problem 8: Finite differences for the 1D Bratu problem
- Problem 9: Integrate functions of finite element expansions
- Problem 10: Finite elements for the 1D Bratu problem
- Problem 11: Derive the Newton system from a variational form
- Problem 12: Derive algebraic equations for nonlinear 1D heat conduction
- Problem 13: Investigate a 1D problem with a continuation method
- Bibliography
