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Ordinary Differential Equations

In: Difference Matrices for ODE and PDE

Author

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  • John M. Neuberger

    (Northern Arizona University, Department of Mathematics and Statistics)

Abstract

Summary In this chapter, we use difference matrices to solve ordinary differential equations. We first apply Newton’s method to a second-order elliptic semilinear boundary value problem. Next, we use MATLAB®’s built-in linear system solver ‘backslash’ to solve a linear ordinary second-order boundary value problem. An eigenvalue problem is solved using second difference matrices, and Fourier Sine series are used to introduce eigenfunction expansion. Next, we consider how to enforce 0-Dirichlet, 0-Neumann, and periodic boundary conditions using both point grid and cell grid. We solve first-order initial value problems, linear systems of first-order linear initial value problems, and first-order nonlinear initial value problems. The difference matrix approach is compared with more traditional ordinary differential equation solvers, e.g., Runge–Kutta and MATLAB’s built-in solver ode45.

Suggested Citation

  • John M. Neuberger, 2023. "Ordinary Differential Equations," Springer Books, in: Difference Matrices for ODE and PDE, chapter 0, pages 63-92, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-12000-8_3
    DOI: 10.1007/978-3-031-12000-8_3
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