Introduction to Partial Differential Equations: A Computational Approach Aslak Tveito Ragnar Winther Springer [...]... order, ordinary differential equations, partial differential equations, and homogeneous and nonhomogeneous equations All these terms can be used to characterize an equation simply by its appearance In this section we will discuss some properties related to the solution of a differential equation 1.2.1 An Ordinary Differential Equation Let us consider a prototypical ordinary differential equation, u (t)... rich and contains a large variety of different species However, there is one basic feature common to all problems defined by a differential equation: the equation relates a function to its derivatives in such a way that the function itself can be determined This is actually quite different from an algebraic equation, say x2 − 2x + 1 = 0, whose solution is usually a number On the other hand, a prototypical... that for this problem, a small change in the initial condition leads to small changes in the solution In fact, the difference between the solutions is reduced at an exponential rate as t increases This property is illustrated in Fig 1.1 4 We will see later that it may also be difficult to check that a certain candidate is in fact a solution This is the case if, for example, the candidate is defined by an... Scene You are embarking on a journey in a jungle called Partial Differential Equations Like any other jungle, it is a wonderful place with interesting sights all around, but there are also certain dangerous spots On your journey, you will need some guidelines and tools, which we will start developing in this introductory chapter 1.1 What Is a Differential Equation? The field of differential equations is... part is, of course, finding the candidate.4 The motivation for studying differential equations is to a very large extent—their prominent use as models of various phenomena Now, if (1.5) is a model of some process, say the density of some population, then u0 is a measure of the initial density Since u0 is a measured quantity, it is only determined to a certain accuracy, and it is therefore important to. .. differential equations into partial differential equations (PDEs) and ordinary differential equations (ODEs) PDEs involve partial derivatives, whereas ODEs only involve derivatives with respect to one variable Typical ordinary differential equations are given by (a) u (t) = u(t), (b) u (t) = u2 (t), (c) u (t) = u(t) + sin(t) cos(t), (1.1) 2 (d) u (x) + u (x) = x , (e) u (x) = sin(x) Here (a) , (b) and (c) are “first... on ideas and basic principles, we shall consider only the simplest possible equations and extra conditions In particular, we will focus on pure Cauchy problems These problems are initial value problems defined on the entire real line By doing this we are able to derive very simple solutions without having to deal with complications related to boundary values We also restrict ourselves to one spatial dimension... t) and the initial condition φ = φ(x) to be given smooth functions.6 As mentioned above, a problem of the form (1.20)–(1.21) is referred to as a Cauchy problem In the problem (1.20)–(1.21), we usually refer to t as the time variable and x as the spatial 6 A smooth function is continuously differentiable as many times as we find necessary When we later discuss properties of the various solutions, we shall... for any constants α and β and any relevant2 functions u and v An equation of the form (1.3) not satisfying (1.4) is nonlinear Let us consider (a) above We have L(u) = u − u, and thus 2 We have to be a bit careful here in order for the expression L(u) to make sense For instance, if we choose u= −1 1 x ≤ 0, x > 0, then u is not differentiable and it is difficult to interpret L(u) Thus we require a certain... | From calculus we know that lim (1 + )1/ = e, →0 so clearly lim E(∆t) = 0, ∆t→0 meaning that we get convergence towards the correct solution at t = 1 In Table 1.1 we have computed E(∆t) and E(∆t)/∆t for several values of ∆t From the table we can observe that E(∆t) ≈ 1.359∆t and thus conclude that the accuracy of our approximation increases as the number of timesteps M increases As mentioned above, . Introduction to Partial Differential Equations: A Computational Approach Aslak Tveito Ragnar Winther Springer