Differential equations. Background mathematics review

Transcription

1 Differential equations Background mathematics review David Miller

2 Differential equations First-order differential equations Background mathematics review David Miller

3 Differential equations in one variable A differential equation involves derivatives of some function E.g., dy dx ay where a is some real number The solution to this equation is y Aexpax where A is an arbitrary constant It is easy to verify solutions but it can be harder to find them

4 First order differential equations dy is a first order differential equation dx ay no derivatives higher than first order The solution y Aexpax has one undetermined constant A and is called a general solution Undetermined constants become fixed using boundary conditions

5 Boundary condition Suppose a ramp has a slope dy 0.4y dx The general solution is y Aexp 0.4x If we also know the boundary condition that at x 0, y 1.5 then, since exp(0)

6 Boundary condition 4 Suppose a ramp has a slope dy 0.4y dx The general solution is y Aexp 0.4x If we also know the boundary condition that at x 0, y 1.5 then, since exp(0) 1 y 1.5exp 0.4x

7 Boundary condition 4 With the same kind of ramp, i.e., dy 0.4y so y Aexp 0.4x dx suppose we know instead that dy at x 0, 0.6 dx dy Since 0.4Aexp0.4x, dx dy then Aexp00.4A dx x

8 Boundary condition 4 With the same kind of ramp, i.e., dy 0.4y so y Aexp 0.4x dx suppose we know instead that dy at x 0, 0.6 dx dy Since 0.4Aexp0.4x, dx dy then Aexp00.4A dx x 0 so A 0.6 /

9 Boundary condition types Boundary conditions that specify the value of a function at some position or boundary are called Dirichlet boundary conditions Boundary conditions that specify the derivative or slope of a function at some position or boundary are called Neumann boundary conditions

10 Imaginary first derivative Note that the equation dy dx iby with b real has the general solution y Aexp ibx A cosbx isin bx Note that neither cosbx nor sinbx is a solution of this equation

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12 Differential equations Second-order differential equations Background mathematics review David Miller

13 Second-order differential equations A second-order differential equation contains no derivatives higher than second order, e.g., first derivatives second derivatives but no higher derivatives such as third order dy dx 2 d y 2 dx 3 d y 3 dx

14 Simple harmonic oscillator equation One simple and important equation is Any of the functions is a solution If we think of t as time these all represent oscillations with angular frequency or frequency f /2 hence the name 2 d y 2 dt 2 y exp( it) exp( it) cos( t) sin( t)

15 General solutions Since this is a second order equation the general solution needs two arbitrary constants Possible general solutions include where A, B, C, D, F, and are arbitrary constants Any of these solutions can be rewritten as any other one they are all equivalent sin y Acos t B t exp exp sin y C i t D i t y F t

16 Notation for time derivatives Since derivatives with respect to time arise in many situations there is a shorthand notation using dots above the variable being differentiated with respect to time with the number of dots indicating the order of differentiation da e.g., a and dt a 2 da 2 dt

17 Helmholtz equation With the same mathematics but with the second derivative in position z instead of time t 2 2 d y 2 d y 2 k zor equivalently kz 2 dz dz which describes oscillations in space rather than time an example of a wave equation Example amplitude of a standing wave on a string 2 0 S

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19 Differential equations Partial differential equations Background mathematics review David Miller

20 Classical one-dimensional scalar wave equation The equation zt, 1 zt, z c t relates how strongly curved a function is (from the second spatial derivative) to the second time (or temporal) derivative We could write it more completely as 2 zt, 1 2 zt, z c t t z but usually we will not bother to do so 0

21 Classical one-dimensional scalar wave equation For this equation 2 zt, 1 2 zt, z c t we can easily check that any functions of the form f z ct or g z ct are solutions E.g., by the chain rule, at some time t o with s z ct o f zcto df s s df s z ds z ds szct o tt o szct s f z ct df s d f s d f s z z ds ds z s z ct s z ct tt ds o szct o o o o

22 Classical one-dimensional scalar wave equation Similarly, at some position z o with s z ct s f zcto df s df s c t ds t ds zz szoct o szoct s c t t ds ds z ds f z ct df s c d f s c d f s sz ct sz ct zzo sz ct o o o o So, at some position z o and time t o f 1 f d f s 1 c d f s z c t ds c ds as required for any f z ct to be a solution 0

23 Wave equation solutions forward waves The difference between the function f z ct 1 and the function f z ct 2 is that f z ct 2 is shifted to the right by ct 2 t1 f z ct 1 f z ct 2 ct t 2 1 The wave moves to the right with velocity c

24 Wave equation solutions backward waves The difference between the function gz ct 1 and the function g z ct 2 is that gz ct 2 is shifted to the left by ct 2 t1 g z ct 2 g z ct 1 ct t 2 1 The wave moves to the left with velocity c

25 Signs and propagation directions It is not the absolute sign of z or ct in f z ct or in gz ct that matters only the relative sign of z and ct f ct z is still a wave going to the right though its shape is the opposite of f z ct f z ct f ct z

26 Monochromatic waves Often we are interested in waves oscillating at one specific (angular) frequency i.e., temporal behavior of the form T t exp( it), exp( it), cos( t), sin( t) or any combination of these 2 2 Then writing z, t Z ztt, we have 2 t leaving a wave equation for the spatial part 2 2 dzz 2 2 kzz0 where k 2 2 dz c the Helmholtz wave equation

27 Wave equations and linearity The wave equation 2 zt, 1 2 zt, z c t and therefore also the Helmholtz equation 2 dzz 2 kzz0 2 dz are linear If 1 z, t and 2 z, t are both solutions so also is any combination a1 z, t b 2 z, t linear superposition

28 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

29 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

30 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

31 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

32 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

33 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

34 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

35 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

36 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

37 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

38 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

39 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

40 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

41 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

42 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

43 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

44 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

45 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

46 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

47 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

48 Standing waves An equal combination of forward and backward waves, e.g., z, t sin kzt sin kzt 2cos t sin kz where k / c gives standing waves E.g., for a rope tied to two walls a distance L apart with k 2 / L and 2 c/ L

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50 Differential equations Laplacian and gradient operators Background mathematics review David Miller

51 Laplacian operator The Laplacian operator can be defined for ordinary Cartesian coordinates x y z A convenient way to say it is del squared 2 Sometimes it is written instead of though we will not use this notation

52 Wave equation in three dimensions We can propose a three-dimensional wave equation 2 xyzt,,, 2 xyzt,,, 2 xyzt,,, 1 xyzt,,, x y z c t which we can write more compactly as 2 1 xyzt,,, xyzt,,, c t or just as c t With some simplifying assumptions it describes many acoustic and electromagnetic waves 0

53 Gradient operator The gradient of a scalar function f( x, y, z) is f f f f grad f i j k x y z For a function hx, y in two dimensions, such as the height of a hill we could define a two-dimensional gradient h h xyh i j x y giving the magnitude and the vector direction of the largest slope

54 Gradient notation Note that 1) though f has no bold font or other vector notation it is a vector quantity 2) we do allow ourselves to put subscripts on it for clarity on how many and what coordinates we are considering as in xy to represent a two-dimensional gradient The symbol can be called del or nabla

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