01246 – Partial Differential Equations
Partial Differential Equations
Resume of lectures by
Ove Skovgaard
© 2005
Edited by
Tore Skogberg
Abstract
The document starts with an introduction to solving partial differential equations and continues with a collection of example problems (the lecture notes). The last two chapters concern the Sturm-Liouville theory and Integral Transformations. Most of the material is using rectangular coordinates although some examples are included with polar, cylindrical and spherical coordinates. There is no big difference although the Laplace operator looks different in the various coordinate systems; however, the process of solving the equations is the same: Reduce the problem to a set of ordinary differential equations, which can be solved using standard methods, and the tool number one in doing so is the separation of variables method.
Ove Skoggaard recommended making notes during his lectures and the present work represents my notes as well as a few additions, which could be valuable for the exam. The lecture notes are introduced with the date of the lecture as (day/month), and my additions are shown without this reference but with reference to the book: Asmar, Nakhlé H. “Partial Differential Equations”, second edition, 2000, Pearson Prentice Hall, ISBN 0-13-148096-0.
The information contained in this document is believed correct, but errors are unavoidable; hence, comments and corrections are welcomed. The document is not intended for distribution.
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Tore Skogberg – Engtoften 14 – 2760 Måløv – 44971344 – -
01246 – Partial Differential Equations
Contents
1Methods
1.1Separation of variables
1.2Ordinary differential equations
1.2.1First order –– y’ + p(x)y = g(x)
1.2.2Second order –– ay’’ + by’ + cy = 0
1.2.3Euler –– r2R’’ + rR’ - 2R = 0
1.2.4Bessel –– x2y’’ + xy’ + (x2 – p2)y = 0
1.2.5Modified Bessel –– x2y’’ + xy’ – (x2 + p2)y = 0
1.2.6Legendre –– (1 – s2)y’’ – 2sy’ + y = 0
1.3Miscellaneous definitions
1.3.1Homogeneous PDE
1.3.2Linear PDE
1.3.3Boundary conditions
1.3.4Chain differentiation
1.3.5Directional derivative
1.3.6Gradient
1.3.7Laplace operator
1.3.8Partial integration
1.4Series expansion
1.4.1Fourier series
1.4.2Half-range Fourier cosine series
1.4.3Half-range Fourier sine series
1.4.4Bessel series
1.4.5Orthogonal function
1.5(5/10) – Non-homogeneous boundary conditions
1.6(5/11) – Non-homogeneous PDE
2(30/8): hux – kuy + qu = s(kx + hy)
3(7/9): uxx – c2uyy = 0
3.1Product solution
3.2ODE with homogeneous BC
3.2.1Case I – Reel roots
3.2.2Case II – One root
3.2.3Case III – Complex conjugate roots
3.3ODE with non-homogeneous BC
4(14/9): cuyy – ux = 0
4.1ODE with homogeneous BC
4.1.1Case I – Real roots
4.1.2Case II – One root
4.1.3Case III – Complex conjugated roots
4.1.4ODE with non-homogeneous BC
4.2Complete solution
5(21/9): 2u = 0
5.1ODE’s with homogeneous BC
5.2ODE with non-homogeneous BC
5.3Complete solution
6(5/10): uxxx – cuyy + u = f(x)
6.1Step 1 – Sub problem for u = u(y)
6.1.1Case I – Real roots
6.1.2Case II – One root
6.1.3Case III – Complex conjugated roots
6.2Step 2 – Generalisation into u = u(x,y)
6.3Step 3 – Generate derivatives
6.4Step 4 – Non-homogeneous problem
6.5Step 5 – Assemble parts
7(12/10): uxxx + uxyy – c(x)uyy + u = f(x)
7.1Step 1 – Sub problem for u = u(y)
7.2Step 2 – Generalisation into u = u(x,y)
7.3Step 3 – Generate derivatives
7.4Step 4 – Non-homogeneous problem
7.5Step 5 – Assemble parts
8(26/10): Curved coordinates
8.1Gradient
8.2Laplace operator
9(26/10): 2u = 0
9.1Boundary conditions
9.2Separation of variables
9.2.1ODE with homogeneous BC
9.2.2ODE with non-homogeneous BC
9.2.3Eigenfunctions
9.3Solution
10(2/11): 2u = 0
10.1Boundary conditions
10.1.1Boundary condition BC1
10.1.2Boundary condition BC2
10.1.3Boundary condition BC3
10.1.4Boundary condition BC4
10.2Separation of variables
10.3Solution to ODE1
10.3.1Case III: Equation
10.3.2Case III: Bessel equation
10.3.3Case III: Solution
10.3.4Case I: Equation
10.3.5Case I: Modified Bessel equation
10.3.6Case I: Solution
10.3.7Case II
10.4Solution to ODE2
10.4.1Bessel orthogonality
10.4.2Solution
11(9/11): 2u = 0
11.1Boundary conditions
11.1.1Boundary condition BC1
11.1.2Boundedness conditions BC2, BC3 and BC4
11.2Separation of variables
11.3Solution to ODE 1
11.3.1Legendre differential equation
11.3.2Solution
11.4Solution to ODE2
11.5Solution to the PDE
12(16-23/11): Sturm-Liouville Theory
12.1Series expansion
12.2Sturm-Liouville form
12.2.1Regular problems
12.2.2Singular problems
12.2.3Example 1
12.2.4Example 2
12.3Transforming into SL-form
12.3.1Example 3
12.3.2Example 4
12.4Results of SL theory
12.4.1Regular SL-problem
12.4.2Singular SL-problem
12.4.3Example 4 continued
13(23-30/11): Integral transformations
13.1Definition
13.1.1Fourier cosine transform
13.1.2Fourier sine transform
13.1.3Fourier transform
13.2Examples
13.2.1Example 1
13.2.2Example 2
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Tore Skogberg – Engtoften 14 – 2760 Måløv – 44971344 – -
01246 – Partial Differential Equations
1Methods
Partial differential equations are solved using different techniques transforming the original problem into a set of ordinal differential equations, or equations that can be treated as ordinary differential equations, and these equations are then solved using the existing set of solution methods. The homogeneous boundary conditions are used to simplify the solutions, such as removing unbounded terms, and the solutions to the ordinary differential equations are assembled into a complete solution set. The non-homogeneous boundary conditions are then used to determine the coefficients using series expansion of orthogonal functions.
Two methods will be used for transformation into ordinary differential equations; separation of variables and integral transformation, of which the first dominates the set of examples presented within this document.
1.1Separation of variables
Many problems can be brought into a condition where the solution can be described by independent differential equations, one for each coordinate of the coordinate system. It may be required reformulating the problem, such as using curved coordinate systems.
Consider as an example the following problem: the one-dimensional wave equation with the independent coordinates x for space and t for time:
The solution is u(x,t) but the example below may equally well represent a similar problem u(x,y) with space coordinates x and y.
The method of separation of variables is as follows:
- Introduce the assumption that the problem can be represented by a product of two (or more) independent functions, one for each of the coordinates. This is equivalent to assume, that the original problem can be described by a set of uncoupled differential equations.
- Differentiate with respect to the independent variables, using the fact that X(x) is independent of t and T(t) is independent of x, so one function is constant while differentiating the other.
- Insert into the PDE, divide by XT and re-arrange the equation to separate the variables onto each side of the equation sign.
- Argue that the ratio T’’/T must be independent of the ratio X’’/X since they are defined through different independent variables, so the ratios must be constant and equal to the same constant; the separation constant, where the minus sign is a convention. Now write the corresponding ordinary differential equations which transforms the original two-dimensional PDE problem into two ordinary differential equations (ODE’s), which can be solved using standard techniques.
- Start with the ODE with homogeneous boundary conditions (in this example assumed to be the x-coordinate) and specify an eigenvalue problem, which defines an infinite set of solutions: 0, 1,2, …, etc. Three cases are considered: < 0, = 0 and > 0 and all the found -values represents valid eigenfunctions. In this example, assume that only the situation with > 0 has useful solutions. The characteristic equation for the X-problem becomes 2 + = 0, so the solutions are given by 2 = –. Introduce the substitution = 2, > 0, which gives = ±i. The solution is:
- Use the homogeneous boundary conditions to determine the constants. If the function value is specified at a point x, then insert the value and solve the equation. If the derivative is specified, then differentiate X(x) and use the condition. As an example, the boundary condition is specified as zero function value at x = 0, thus u(0,t) = 0, which means that X(0)T(t) = 0, which implies that X(0) = 0, since T(t) is independent upon x. The example gives A = 0 and thus X(x) = Bsin(x). As an example, the other boundary condition is specified as u(b,t) = 0; the relation becomes Bsin(b) = 0, which can be fulfilled by a non-trivial solution (non-zero B) only for b = n, n = 1, 2, 3, … The solution with n = 0 is trivial since X(x) would become identically zero. We have now defined the following eigenfunctions and values of for the present example:
- Now consider the coordinate with non-homogeneous boundary conditions, in this example the T-problem. Since is known at this point, we can re-write the ODE. In the example we have equivalent equations and end up with a solution, which is similar to the X-problem, so the solution is again a set of cosine and sine terms:
- Use any available homogeneous boundary condition to determine constants Cn or Dn. Then proceed as shown below. Do not use non-homogeneous boundary conditions at this point.
- Combine the results obtained so far. The product of constants BnCn and BnDn can be simplified by setting Bn = 1, which does not affect the solution. Any solutions from the cases with < 0 or = 0 are added to u(x,t) before the summation. In this example, solutions are only specified for > 0.
- Use the non-homogeneous boundary conditions to determine the constants Cn and Dn. As an example, the initial condition is specified as u(x,0) = f(x), and we can proceed as follows using the knowledge that the trigonometric functions are orthogonal. Start by multiplying both sides with sin(mx/b). Then integrate both sides through the definition range for the current problem, which in the example is set to 0xb. Then conclude, that the integral is zero for all values of m with exception of m = n. The constant Cn can then be determined.
- Use another non-homogeneous boundary condition to determine constant Dn using a similar technique.
The PDE-problem has been solved.
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01246 – Partial Differential Equations
1.2Ordinary differential equations
A collection of differential equations are listed below. The list is not exhaustive.
1.2.1First order –– y’ + p(x)y = g(x)
The general form of a first order differential equation is (after Asmar page A2):
The functions p(x) and g(x) must be continuous on the definition interval I. The complete solution to the differential equation is:
Constant C is the integration constant. For g(x) = 0, i.e. a homogeneous differential equation: y’ + p(x)y = 0, the solution reduces to:
For p(x) = where is a constant, and g(x) = 0, i.e. a homogeneous equation with constant coefficient: y’ + y = 0, the solution reduces further to:
1.2.2Second order –– ay’’ + by’ + cy = 0
The general form of a second-order differential equation with constant coefficients, which is often labelled the damped harmonic oscillator, is (after Asmar page A16):
It can be shown that the exponential function (with real or complex argument) is a solution and the roots of the characteristic equation determines the arguments to the exponential.
The solutions to the characteristic equation are:
Case I:Two real roots (1,2 = ± ) leads to a solution with exponentials. Two possibilities:
Case II:One root (1,2 = ) leads to the following solution, which is a straight line for 1 = 0:
Case III:Two complex conjugated roots (1,2 = ± i); is the harmonic oscillator solution:
1.2.2.1Example
Given the following eigenvalue problem:
The characteristic equation is with the solutions , where can take any value so three cases must be analysed: Case I: For < 0 substitute = –2, > 0 for = ±, the solution is the sum of two exponentials; use the hyperbolic functions for a limited range. Case II: For = 0 the solution is a straight line. Case III: For > 0 substitute = 2, > 0 for = ±i, the solution is the sum of a cosine and sine functions.
Table 1 – Template functions for the second-order ordinal differential equation: .
Range for / Substitution / Use equation to determine A, B / Comments/ (None) /
Determine the constants from the boundary conditions, such as ux(0,t), u(a,t) or similar, and establish conditions for or (such as = n). Determine the values as: 0, 1, 2, …, or whatever numbering is appropriate. Remember to exclude the trivial null-solution, such as ; i.e. be careful to start numbering as appropriate, such as n = 0, 1, 2, ... or n = 1, 2, 3, ... as required.
Assemble the eigenvalue functions from the analysis:
The solutions are assembled from the following collection of differential equations, as required by the problem at hand.
1.2.3Euler –– r2R’’ + rR’ - 2R = 0
The Euler differential equation is a typical result of separation of variables with the function substituted by u = R(r)() or similar, and is for the R component defined as:
The coefficients are not constant, so the conventional solution approach does not apply. Note that the exponent to r is identical to the differentiation order of R.
The solution is found by assuming a solution type of:
Differentiation generates the following expressions for the derivative of R:
Insertion into the ordinary differential equation for R results in:
Hence the following solution to the problem in r:
Use the boundary conditions to determine An and Bn. Typically the boundedness condition leads to Bn = 0 to limit the solution for r 0.
1.2.4Bessel –– x2y’’ + xy’ + (x2 – p2)y = 0
Bessel (1784-1846) studied the following equation, which is known as the Bessel equation of order p (Asmar page 237):
The equation is of second order so there will be two independent solutions to the equation and the complete solution is the sum of the two functions. The solutions are called the Bessel function of the first kind and order p, Jp(x), and the Bessel function of the second kind and order p, Yp(x). The solution is:
The Bessel functions Jp(x) are bounded while Yp(x) are not bounded for x approaching zero. All functions approach zero for x approaching infinity and oscillates around the x-axis. The first of the Bessel function of the first kind starts from J0(0) = 1 while the higher orders starts from Jp(0) = 0 with a slope of p.
Figure 1 – Bessel functions of the first kind and order zero (left) and the modified Bessel functions of the second kind and order zero (right). See also Asmar page 212, 229 and 240.
The zero-crossing values of the Bessel functions are called p1, p2,p3, ..., and are found in tables, such as Asmar page 250 for curve of Jp(x) zeros and 251 for table of Jp(x) zeros.
1.2.5Modified Bessel –– x2y’’ + xy’ – (x2 + p2)y = 0
The general form of the modified Bessel differential equation is:
The equation is of second order so there will be two independent solutions to the equation and the complete solution is the sum of the two functions. The solutions are called the modified Bessel function of the first kind and order p, Ip(x), and the modified Bessel function of the second kind and order p, Kp(x). The solution is:
The functions are basically exponential and corresponds roughly to the hyperbolic functions. Both functions are un-bounded; Ip approach infinity for x approaching infinity and Kp approach infinity for x approaching zero.
1.2.6Legendre –– (1 – s2)y’’ – 2sy’ + y = 0
Solutions to the Legendre differential equation exists only for = 0, 2, 6, 12, … (from Asmar page 272 and 282):
The solution is the associated Legendre function:
The function is a polynomial, which is non-zero for 0 mn.
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01246 – Partial Differential Equations
1.3Miscellaneous definitions
1.3.1Homogeneous PDE
A homogeneous partial differential equation consists of terms with the function value u, or the derivatives, such as ux, uxx, uy, uyy, …, ut, … A non-homogeneous partial differential equation consists of the above terms plus other terms. Examples:
1.3.2Linear PDE
The PDE is linear if the dependant variable u and all partial derivatives are of the first degree (power of one) and at most one of these appears in any given term:
Otherwise it is non-linear:
1.3.3Boundary conditions
Boundary conditions are used to determine the integration constants from the solution process. A boundary condition can be of three different types, Dirichlet, Neumann or Robin and are defined as follows (Asmar page 180-181):
- A Dirichlet boundary condition is any condition that specifies the values of the solution u on the boundary.
- A Neumann boundary condition is any condition that specifies the normal derivative on the boundary.
- A Robin condition is any combination that specifies both the value of the solution and the normal derivative on the boundary.
1.3.4Chain differentiation
Let function f(x) and the derivative df(x)/dx be continuous on the open interval axb. If further x = x(t) and dx/dt are continuous functions of variable t over the interval t, such that x(t) is in the open interval ax(t) < b, we speak of F(t) = f(x(t)) = f(x) as a composite function of t. The derivative of the function F(t) with respect to time is given by the chain rule (from 01246 preamble, page 5):
1.3.5Directional derivative
Let u(x,y) be a scalar function with continuous first order spatial derivative in an open spatial domain D and let be an arbitrary spatial unit vector. The directional derivative dy/ds at any point P in domain D, in the direction of an arc length coordinate s through P (with direction and scale of length of s defined by the vector ) is (after 01246 preamble, page 7):
This gives the rate of change of u at any point P in any direction e.
1.3.6Gradient
The gradient of u (gradu or u) where u is a scalar function with continuous first order spatial derivatives in an open spatial domain D, is (after 01246 preamble, page 6):
See Asmar for definition of Polar coordinates page 194, Cylindrical coordinates page 196 and Spherical coordinates page 197.
1.3.7Laplace operator
The Laplace operator is defined as (Asmar page 194-197):
1.3.8Partial integration
Partial integration of the product of two functions f(x) and g(x) is defined as:
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Tore Skogberg – Engtoften 14 – 2760 Måløv – 44971344 – -
01246 – Partial Differential Equations
1.4Series expansion
A function is represented by a sum of terms using an orthogonal function.
1.4.1Fourier series
Let function f(x) be 2-periodic, i.e. f(x) = f(x + 2). The function can be expressed by:
For use within a PDE-problem only the range within the open interval –x is of concern. The periodicity is of no importance, since the PDE-problem is not defined outside the interval. Note that the function f(x) is not defined at the endpoints. The expressions can be defined for another 2-interval, such as 0 < x < 2. See also the Sturm-Liouville theory.
1.4.2Half-range Fourier cosine series
The half-range cosine series is defined as:
1.4.3Half-range Fourier sine series
The half-range sine series is defined as:
The cosine series is an even expansion (symmetrical around x = 0), while the sine series is an odd expansion (symmetrical around (0,0), the origin). Both are valid for any function f(x) but requires different numbers of terms to converge. Note that the series expansion is defined within an open interval – the expansion is not valid at the endpoints of the interval.
1.4.4Bessel series
The series expansion using the Bessel function of the first kind and order 0, for the finite interval 0 < xa, is defined as (see example 9/11):
1.4.5Orthogonal function
A function is orthogonal on the interval from a to b, if:
The definition of an orthogonal function for Sturm-Liouville problems, is:
Function w(x) is a weight function.
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Tore Skogberg – Engtoften 14 – 2760 Måløv – 44971344 – -
01246 – Partial Differential Equations
1.5(5/10) – Non-homogeneous boundary conditions
As an introduction to the method of dividing a complex PDE problem into less-complex PDE problems, consider the following linear and homogeneous PDE problem with four non-homogeneous boundary conditions:
The PDE problem can be transformed into two PDE problems each with two homogeneous and two non-homogeneous boundary conditions by the transformation: