MA 201: PDE Lecture 5 Introduction to Second Order PDE Reference: S. Sankara Rao, Chapter 1 (Section 1.2) (PDE:Lecture-5) MA201(2023) 1 / 23 Second order PDE Second order PDE • Physical problems are governed by many different partial differential equations. A few problems are governed by a single first-order PDE. Numerous problems are governed by higher order PDEs. • Solutions of PDEs of order higher than the first exhibit distinctive types of behaviour. • Physical problems fall into one of the following three general classifications: 1 2 3 Propagation problems: The solution u(x, t) tends to propagate with some coherence. Diffusion problems: The solution u(x, t) tends to smear out signals. Steady-state problems: The solution here is in steady-state and do not vary in time at all. (PDE:Lecture-5) MA201(2023) 2 / 23 Second order PDE Propagation Problems Wave Equation • The simplest situation to give rise to a one-dimensional wave equation is the motion of a stretched string- specifically the transverse vibrations of a string such as the string of a musical instrument. • Assume that a string is placed along the x-axis, and then fixed at ends x = 0 and x = L; it is then deflected and at some instant, which we call t = 0, is released and allowed to vibrate. • The quantity of interest is the deflection u of the string at any point x, 0 ≤ x ≤ L and at any time t > 0. • We write u = u(x, t). (PDE:Lecture-5) MA201(2023) 3 / 23 Second order PDE The following diagram shows a possible displacement of the string at a fixed time t. u(x,t) 0 x L Figure: x versus u(x,t) (PDE:Lecture-5) MA201(2023) 4 / 23 Second order PDE Subject to various assumptions... 1. neglecting damping forces such as air resistance. 2. neglecting the weight of the string. 3. assuming that the tension in the string is tangential to the curve of the string at any time. 4. assuming that the string performed small transverse oscillations i.e., every particle of the string moves strictly vertically and such that its deflection and slope s every point on the string is small. ..., it can be shown, by applying Newton’s Law of motion to a small segment of the string, that u satisfies the PDE ∂2u ∂2u = c2 2 2 ∂t ∂x (1) where c 2 = Tρ , ρ being the mass per unit length and T being the horizontal component of the tension in the string. (PDE:Lecture-5) MA201(2023) 5 / 23 Second order PDE • To determine u(x, t) uniquely, we must also know (a) the initial displacement of the string at the time t = 0 at which it is released. (b) the initial velocity of the string. • Thus we must be given u(x, 0) = f (x)( Initial position) ∂u (x, 0) = g (x)( Initial velocity) ∂t The PDE (1) is the (undamped) wave equation. We will discuss solution of it for various initial conditions. We also consider the following wave equations which arise under different assumptions: (a) (b) ∂2u ∂t 2 ∂2u ∂t 2 2 = c 2 ∂∂xu2 − g if the weight of the string is also taken into consideration. 2 = c 2 ∂∂xu2 − α ∂u ∂t if a damping force proportional to the velocity of the string (with a damping constant α) is included. (PDE:Lecture-5) MA201(2023) 6 / 23 Second order PDE Diffusion Problems The Heat Conduction Equation • Consider a long thin bar, or wire, of length L and constant cross-section and of homogeneous material oriented along x-axis. • Imagine that the bar is thermally insulated laterally and is sufficiently thin that heat flows (by conduction) only in x-direction. • Then the temperature u at any point in the bar depends only on the x-coordinate of the point and the time t. x x=0 (PDE:Lecture-5) Heat conduction in a thin rod of length L MA201(2023) x=L 7 / 23 Second order PDE • By applying the principle of conservation of energy, it can be shown that the temperature at point x and at time t, given by u(x, t), satisfies the following PDE: ∂2u ∂u = k 2 , 0 < x < L, t > 0. (2) ∂t ∂x • where k is a positive constant. In fact k, also known as thermal diffusivity of the bar, is given by κ k= sρ where κ = thermal conductivity of the material of the bar, s = specific heat capacity of the material of the bar and ρ = density of the material of the bar. The PDE (2) is called the one dimensional heat conduction equations (or, in other context where it arises, diffusion equation). (PDE:Lecture-5) MA201(2023) 8 / 23 Second order PDE • The fact that (2) is first order in time variable t means that only one initial condition at t = 0 is needed, together with two boundary conditions, to obtain a unique solution. • The usual initial condition u(x, 0) = f (x), where f (x) is a known function, specifies the initial temperature distribution in the bar. • Various types of boundary conditions at x = 0 and x = L is possible. For example, (a) u(0, t) = T1 and u(L, t) = T2 (i.e., the ends of the bar are at constant temperatures T1 and T2 ) (0, t) = 0 = ∂u (L, t) which are insulation condition as there is no heat flow (b) ∂u ∂x ∂x through the ends of the bar. (PDE:Lecture-5) MA201(2023) 9 / 23 Second order PDE Two dimensional heat conduction problem • Consider a thin plate, insulated on its top and bottom surfaces so that no heat flow occurs other than Oxy plane as shown in the figure. • The two dimensional heat conduction problem in the plate is governed by the following equation: ∂u(x, y , t) =k ∂t ∂ 2 u(x, y , t) ∂ 2 u(x, y , t) + 2 ∂x ∂y 2 , 0 < x < a, 0 < y < b, ; t > 0. (3) • Here u(x, y , t) represents the temperature on the plate at the point (x, y ) and at the time t. y y=b x 0 (PDE:Lecture-5) x=a MA201(2023) 10 / 23 Second order PDE • Boundary conditions and initial condition are needed to determine the solution of (3) uniquely. • A typical boundary condition might be: u(x, 0, t) = T1 , 0 ≤ x ≤ a( bottom side is kept at a constant temperature ) ∂u (a, y , t) = 0, 0 ≤ y ≤ b( the right hand side is being insulated ) ∂x u(x, b, t) = T2 , 0 ≤ x ≤ a( top side is kept at a constant temperature ) u(0, y , t) = 0, 0 ≤ y ≤ b( the left hand side is kept at zero temperature) • An initial condition would be of the form u(x, y , 0) = f (x, y ) where f is a given function. y u(x,b,t)=T2 y=b ux(a,y,t)=0 u(0,y,t)=0 x 0 (PDE:Lecture-5) u(x,0,t)=T1 MA201(2023) x=a 11 / 23 Second order PDE Steady state problems Laplace’s equation • Recall the two dimensional heat conduction problem: ∂u(x, y , t) =k ∂t ∂ 2 u(x, y , t) ∂ 2 u(x, y , t) + 2 ∂x ∂y 2 , 0 < x < a, 0 < y < b, ; t > 0. • If the heat conduction is steady, i.e., ∂u ∂t = 0, then the temperature distribution across the plate is independent of the time t and hence u solves the equation ∂2u ∂2u + = 0, 0 < x < a, 0 < y < b. (4) ∂x 2 ∂y 2 • (4) is known as Laplace’s equation. (PDE:Lecture-5) MA201(2023) 12 / 23 Second order PDE Both equation (4) and its three-dimensional analogous form, viz., ∂2u ∂2u ∂2u + 2 + 2 = 0, 0 < x < a, 0 < y < b, 0 < z < c, 2 ∂x ∂y ∂z (5) arise in wide variety of applications, quite apart from the steady state of heat conduction theory. Since time does not arise in (8) or (9) it is evident that Laplace’s equation is always a model for equilibrium situations. Some areas in which Laplace’s equation arises are (a) electrostatics (u being the electrostatic potential in a charge free region) (b) gravitation (u being the gravitational potential in free space) (c) steady state flow of inviscid fluids (d) steady state heat conduction. In any problem involving Laplace’s equation, we are interested in solving it in a specific region R for given boundary conditions. (PDE:Lecture-5) MA201(2023) 13 / 23 Second order PDE Boundary Condition • The boundary condition is a set of constraints that define the behavior of unknown functions on the spatial boundary of the domain. • A PDE with a boundary condition is also called a boundary value problem. • Four main types of boundary conditions exist: (i) Dirichlet Condition: The Dirichlet condition specifies the values of unknown function u on the boundary of the domain. For example, for the problem in a finite one-dimensional medium of length L, then the Dirichlet conditions are u(0, t) = T1 (t) and u(L, t) = T2 (t). (ii) Neumann Condition: The Neumann condition specifies the values of the normal derivative on the boundary of the domain. For example, ux (0, t) = T1 (t) and ux (L, t) = T2 (t). (PDE:Lecture-5) MA201(2023) 14 / 23 Second order PDE Boundary Condition (iii) Robin Condition: The Robin condition is a linear combination of the Dirichlet and Neumann conditions. For example, a1 u(0, t) + a2 ux (0, t) = T1 (t) and c1 u(L, t) + c2 ux (L, t) = T2 (t). (iv) Mixed Condition: This condition occurs when Dirichlet condition is specified on some part of the boundary and Neumann condition is specified on the remaining part of the boundary. For example, u(0, t) = T1 (t), ux (L, t) = T2 (t). (PDE:Lecture-5) MA201(2023) 15 / 23 Second order PDE Initial Condition For an evolutionary( or time-dependent) equation, a boundary condition does not guarantee a unique solution. We also need to define the initial condition, which specifies the value of unknown function u(x, t) at the initial time on the domain. This is called the Cauchy initial condition. For example, u(x, 0) = f (x). If the evolutionary equation contains the higher order derivatives with respect to time t, we need to specify the additional values of derivatives up to the highest order of the derivative of t. (PDE:Lecture-5) MA201(2023) 16 / 23 Classification of PDEs Classification of PDEs • An ODE is classified according to its order and whether it is linear or nonlinear. • PDE models are more difficult to classify. • Order and linearity are not the only issues. • In fact the structure of the PDE dictates what type of boundary and initial conditions are to be imposed. (PDE:Lecture-5) MA201(2023) 17 / 23 Classification of PDEs Classification of PDEs Consider the following second-order equations: ut − αuxx = 0 (Heat Equation) utt − c 2 uxx = 0 (Wave Equation) uxx + uyy = 0 (Laplace’s Equation). • The first two equations are evolution equations that describe how a process evolves in time. • We expect that each would require initial data, that is, the state of the system at time t = 0. • The third one is an equilibrium equation where time is not an issue, but we expect that boundary conditions are appropriate. • The solutions of the equations are expected to behave differently. (PDE:Lecture-5) MA201(2023) 18 / 23 Classification of PDEs Classification of Second order PDEs If physical classifications are allowed, we would call these equations diffusive, wave-like and static, respectively. We observe • Diffusion tends to smear out signals. • Waves tend to propagate with some coherence. • Solutions to Laplace’s equation are steady-state and do not vary in time at all. (PDE:Lecture-5) MA201(2023) 19 / 23 Classification of PDEs Classification of Second order PDEs General second-order semi-linear partial differential equation in two independent variables x, t is of the form Auxx + Buxt + Cutt + Dux + Eut + F (x, t, u) = 0, (6) where A, B, C , D and E are smooth functions of x and t. • Second-order derivatives appear linearly. • Lu ≡ Auxx + Buxt + Cutt is called the principal part of the equation. • Let assume that A, B, C in (6) are constants and A ΜΈ= 0. Then the differential operator in the principal part of (6) can be also written as ∂2 B ∂ ∂ C ∂2 ∂ ∂ ∂ ∂ + + = − ω − ω 1 2 ∂x 2 A ∂x ∂t A ∂t 2 ∂x ∂t ∂x ∂t (PDE:Lecture-5) MA201(2023) (7) 20 / 23 Classification of PDEs • where ω1 , ω2 are defined as (on comparing both sides) B C ω1 + ω2 = − , ω1 ω2 = A A . • Equivalently ω1 , ω2 are the roots of the quadratic equation Aω 2 + Bω + C = 0. (8) • Since we are generally interested in obtaining real valued solutions of (6), the possibility of using the factorization (7) to simplify (6) depends on whether ω1 , ω2 are real or complex valued. • This is determined by the sign of the discriminant D = B 2 − 4AC of the quadratic equation (8). (PDE:Lecture-5) MA201(2023) 21 / 23 Classification of PDEs Classification of Second order PDEs (Contd.) • Thus we can classify second order semi-linear equation with constant coefficients based on the sign of the quantity D = B 2 − 4AC , called the discriminant. • This approach of classification also hold in the case when A, B, C are functions of x and t where the sign of the discriminant at a point depends on the values of x and t at that point. • Equation (1) is called hyperbolic at (x, t) if D > 0 at (x, t). • Equation (1) is called parabolic at (x, t) if D = 0 at (x, t). • Equation (1) is called elliptic at (x, t) if D < 0 at (x, t). (PDE:Lecture-5) MA201(2023) 22 / 23 Classification of PDEs Example: Under this classification • The wave equation: utt − c 2 uxx = 0, is hyperbolic. • The diffusion or heat equation: ut − αuxx = 0 is parabolic. • The Laplace’s equation: uxx + uyy = 0 is elliptic. • The Tricomi equation: uyy − yuxx = 0 is elliptic if y < 0, parabolic if y = 0, and hyperbolic if y > 0. For quasi-linear second order PDEs, a similar classification can be made, but the resultant type (hyperbolic, parabolic, or elliptic) may now depend on the particular solution u being considered. For example, uxx − uuyy = 0 is elliptic if u < 0 and hyperbolic if u > 0. (PDE:Lecture-5) MA201(2023) 23 / 23