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You gain access to the e-book versions of our study guide and all of our forms, including the Actual Exam (coming soon). This blog post is an introduction to a series of posts on solving differential equations. The posts will be focused on Self-Adjoint Boundary Value Problems for Linear Hyperbolic Equations, which includes methods for "local" and "global" perturbations. In each post we will create a Linear Hyperbolic equations from Physics, Physical chemistry or Material science problems so that you can apply what you learn from this post. We will also teach you how to solve them using these methods. This is the first post in the series. This post will introduce you to solving Linear Hyperbolic Equations using "local" perturbation methods all while knowing that the solutions are globally Lipschitz continuous. The method of Choosing a Method for solving Linear Hyperbolic Equations is an Introduction to Self-Adjoint Boundary Value Problems for Linear Hyperbolic Equations . The method of Choosing a Method for Solving Linear Hyperbolic Equations is where you learn how to solve different kinds of Linear Hyperbolic equations using "local" perturbation methods. To see what all of this is about, let's start with an example problem. Example Problem 1: Solving the Boundary Value Problem for the Space-Time Wave Equation (using local perturbation methods) The following is the contour plot of the solution to this problem. We've chosen to represent solutions by lines (rather than plots). For our convenience, we've also labeled the axes on the plot. The solution is pretty close to a straight line. However, you can see that it has some curvature but not enough that it looks like a hyperbola at all times.

This method applies only when you have continuous solutions in your domain space for your linear hyperbolic equations parameterized by . This is different from the domain space being 0, or another "special" number. This means that is always continuous throughout both dimensions of your domain space, which is good. It also means that the solution to your linear hyperbolic equation can have some curvature in one axis. However, it is important to remember that the solutions are always Lipschitz continuous in both axes. Before You Get Started : You will need to You'll also need to read Chapter 1 . Let's start by working through an example of solving the problem of finding the solution to the boundary value problem for the Space-Time Wave Equation. Our goal will be to find . To do this, we'll use local perturbation methods to create perturbations in our equations that follow a specific hyperbolic trend. Some examples of how we can do this are shown below. Example Problem 2: Solving Boundary Value Problems for Linear Hyperbolic Equations (using "local" perturbation methods) The following is the contour plot of these solutions. You can see that these solutions are very close to straight lines (in fact they are straight lines).

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