What Are Partial Differential Equations?
Let us begin by delineating our field of study. A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables.
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What Are Partial Differential Equations?
Let us begin by delineating our field of study. A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables. For example, d4 u d2 u + 2 + u2 = cos x (1.1) dx4 dx is a differential equation for the function u(x) depending on a single variable x, while ∂u ∂2u ∂2u = + 2 −u ∂t ∂x2 ∂y
(1.2)
is a differential equation involving a function u(t, x, y) of three variables. A differential equation is called ordinary if the function u depends on only a single variable, and partial if it depends on more than one variable. Usually (but not quite always) the dependence of u can be inferred from the derivatives that appear in the differential equation. The order of a differential equation is that of the highest-order derivative that appears in the equation. Thus, (1.1) is a fourth-order ordinary differential equation, while (1.2) is a second-order partial differential equation. Remark : A differential equation has order 0 if it contains no derivatives of the function u. These are more properly treated as algebraic equations,† which, while of great interest in their own right, are not the subject of this text. To be a bona fide differential equation, it must contain at least one derivative of u, and hence have order ≥ 1. There are two common notations for partial derivatives, and we shall employ them interchangeably. The first, used in (1.1) and (1.2), is the familiar Leibniz notation that employs a d to denote ordinary derivatives of functions of a single variable, and the ∂ symbol (usually also pronounced “dee”) for partial derivatives of functions of more than one variable. An alternative, more compact notation employs subscripts to indicate partial derivatives. For example, ut represents ∂u/∂t, while uxx is used for ∂ 2 u/∂x2 , and ∂ 3 u/∂x2 ∂y for uxxy . Thus, in subscript notation, the partial differential equation (1.2) is written ut = uxx + uyy − u. (1.3) † Here, the term “algebraic equation” is used only to distinguish such equations from true “differential equations”. It does not mean that the defining functions are necessarily algebraic, e.g., polynomials. For example, the transcendental equation tan u = u, which appears later in (4.50), is still regarded as an algebraic equation in this book.
P.J. Olver, Introduction to Partial Differential Equations, Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-02099-0_1, © Springer International Publishing Switzerland 2014
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1 What Are Partial Differential Equations?
We will similarly abbreviate partial differential operators, sometimes writing ∂/∂x as ∂x , while ∂ 2 /∂x2 can be written as either ∂x2 or ∂xx , and ∂ 3 /∂x2 ∂y becomes ∂xxy = ∂x2 ∂y . It is worth pointing out that the preponderance of differential equations arising in applications, in science, in engineering, and within mathematics itself are of either first or second order, with the latter being by far the most prevalent. Third-order equations arise when modeling waves in dispersive media, e.g., water waves or p
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