SOLUTION BY THE METHOD OF SEPARATION OF VARIABLES USING FOURIER
Introduction Separation of variables is a solution method for partial differential equations. While its beginnings date back to work of Daniel Bernoulli (1753), Lagrange (1759), and d' Alembert (1763) on wave motion, it is commonly associated with the name of Fourier (1822), who developed it for his research on conductive heat transfer. Since Fourier's time it has been an integral part of engineering mathematics, and in spite of its limited applicability and heavy competition from numerical methods for partial differential equations, it remains a well-known and widely used technique in applied mathematics. Separation of variables is commonly considered an analytic solution method that yields the solution of certain partial differential equations in terms of an infinite series such as a Fourier series. While it may be straightforward to write formally the series solution, the question in what sense it solves the problem is not readily answered without recourse to abstract mathematical analysis. A modern treatment focusing in part on the theoretical underpinnings of the method and employing the language and concepts of Hilbert spaces to analyze the infinite series may be found in the text of MacCluer. For many problems the formal series can be shown to represent an analytic solution of the differential equation. As a tool of analysis, however, separation of variables with its infinite series solutions is not needed. Other mathematical methods exist which guarantee the existence and uniqueness of a solution of the problem under much more general conditions than those required for the applicability of the method of separation of variables.
Aims and Objectives The aim of this project was to get acquainted with the separation of variables technique to solve Partial Differential Equations and to apply this method to solve different problems encountered in our day to day life.
Outline of the Method The method of separation of variables is a way of finding particular and general solutions of m ain idea is to consider the additive certain types of partial differential equations (PDE’s). Its main ansatz, or the multiplicative ansatz for a solution of a PDE that allows for reducing this PDE to a set of (uncoupled) ordinary differential equations (ODE’s) for the unknown functions of one variable. variable. Locally, the additive ansatz is, through the change of variables u(x) =
, equivalent to the multiplicative ansatz. Many 1
well known equations of mathematical physics such as the heat equation, the wave equation, the Schrödinger equation and the Hamilton-Jacobi (H.J) equation are solved by separating variables in suitably chosen systems of coordinates. The Solution of variable method can be attributed to J. Fourier, who solved the heat equation
∂t u = ∂ xxu
…………………………………………………………………………………………………………… (A)
for distribution of temperature u(x, t) in a one-dimensional metal rod (of length L) by looking first for special solutions of the product type u(x,t) = X(x)T(t). This ansatz, substituted to (A), 2 2 reduces it to two ODE’s: ∂t T = -k T and ∂ xx X = -k X that can be solved by quadrature’s q uadrature’s::
∑
Due to linearity of (1) any formal linear combination u(x, t) = k X k k (x)T k k (t) is again a solution of the heat equation and can 1x used for solving an initial boundary value problem (IBVP). For instance in the case of the 1BVP on the interval 0 ≤ x ≤ L L and with zero boundary conditions:
only a countable set of values for the separation constant k is admissible: k n = Then the general solution has the form of the Fourier series
where the coefficients c are given by the integral
n = 1,2,…… 1,2,…… The sequence of
functions sin( k n x) is complete on the interval [0, L]. That means that any regular (continuous and differentiable) initial data function f(x) such that f(0) = f(L) = 0 can be uniquely expressed as an infinite convergent sum of the orthogonal set of functions sin(k n x). The study of mathematical properties of the Fourier expansion gave rise to the classical theory of Fourier series and of Fourier integrals. The separation of variables method involves certain steps which can be conveniently followed to achieve the solution. These steps are enlisted below
Separate the variables Assume, for example, that u(x, t) = X(x)T(t). Substitute this into the PDE to get 2 ODE’s for X X and T separately. Decide on the sign of the separation constant The constant arises when you separate the variables. We will solve these equations on the basis of the sign of the separation constant. Solve the separated ODE’s We get, for example, ODE’s to solve for X(x) and T(t) that depend on the constant in Step 2.
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Solve the (homogeneous) boundary conditions, so that we know what X(t) and T(t) are, and reconstruct the function, for example u(x, t) that we need, using u(x, t) = X(x)T(t). Check that the u(x, t) that we have actually solves the problem.
Applying the method to practical problems The method of separation of variables is very helpful in solving different problems. The model of a vibrating elastic string (a violin string, for instance) consists of the one dimensional wave equation. ………………………………………………………….. (1)
2
c = T/ρ
for the unknown deflection u (x, t) of the string, a PDE that we have just obtained, and some additional conditions, which we shall now derive. Since the string is fastened at the ends x = 0 and x = L (see Sec.12.2), we have the two boundary conditions. ……………………….. (2)
Furthermore, the form of the motion of the string will depend on its initial deflection (deflection at time t = 0), call it f(x) and on its initial velocity (velocity at t = 0), call it g(x). We thus have the two initial conditions ……………………….. (3)
We now have to find a solution of the PDE (1) satisfying the conditions (2) and where ut = ∂u/∂t. We (3). This will be the solution of our problem. We shall do this in three steps, as follows. Step 1. By the “method of separating variables” or product method, setting u(x, t) = F(x)G(t), we obtain from (1) two ODEs, one for and the other one for G(t). Step 2. We determine solutions of these ODEs that satisfy the boundary conditions (2). Step 3. Finally, using Fourier series, we compose the solutions found in Step 2 to obtain a solution of (1) satisfying both (2) and (3), that is, the solution of our model of the vibrating string.
Step 1. Two ODEs from the Wave Equation (1)
In the method of separating variables, or product method, we determine solutions of the wave equation (1) of the form u(x, t) = F(x)G(t)
……………………………………………………………………………………………………………………… (4) 3
which are a product of two functions, each depending on only one of the variables x and t . This is a powerful general method that has various applications in engineering mathematics, as we shall see in this chapter. Differentiating (4), we obtain and
where dots denote derivatives with respect to t and primes derivatives with respect to x. By inserting this into the wave equation (1) we have
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Dividing by c FG and simplifying gives
The variables are now separated, the left side depending only on t and the right side only on x. Hence both sides must be constant because, if they were variable, then changing t or x would affect only one side, leaving the other unaltered. Thus, say,
Multiplying by the denominators gives immediately two ODEs ……………………………………………………………………………………………………………………. (5)
and ………………………………………………………………………………………………………………… (6)
Here, the separation constant k is still arbitrary.
Step 2. Satisfying the Boundary Conditions Cond itions (2)
We now determine solutions F and G of (5) and (6) so that u = FG satisfies the boundary conditions (2), that is, u (0, t ) = F (0)G (t ) = 0,
u ( L L, t ) = F ( L L)G (t ) = 0 for all t .
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……………………………………… (7)
We first solve (5). If G = 0, then u = FG = 0, which is of no interest. Hence G ≠ 0 and then by (7), (a) F(0) = 0,
(b) F(L) = 0
……………………………………… (8)
We show that k must be negative. For k = 0 the general solution of (5) is F = ax + b and from (8) we obtain a = b = 0, so that F = 0 and u = FG ≡ 0, which is of no interest. 2
For positive K = μ a general solution of (5) is μ x
F = Ae
μ x
- Be
and from (8) we obtain F ≡ 0 as before. Hence we are left with the possibility of choosing k 2 2 negative, say, k = -p . Then (5) becomes F ˝ + p F = 0 and has as a general solution F ( x x) = A cos px + B sin px.
From this and (8) we have F (0) = A = 0 and then F ( L L) = B sin pL = 0.
We must take B take B ≠ 0 since otherwise F ≡ 0. Hence sin pL = 0. Thus pL = np, so that p =
, where n is an integer
………………………………………………………………….. (9)
Setting B 1, we thus obtain infinitely many solutions F ( x x) = Fn ( x x), where =
, (n = 1,2, … ).
………………………………………………………………………………… (10)
These solutions satisfy (8). [For negative integer n we obtain essentially the same solutions, except for a minus sign, because sin (- α) = -sin α 2
2
We now solve (6) with k = -p = - (nπ /L) resulting from (9), that is, ………………………………………………………………….. (11*)
A general solution is
Hence solutions of (1) satisfying (2) are un( x x, t ) = F n( x x)Gn(t ) = Gn(t )F n( x x), written out (n = 1,2, …).
…………….………………… (11)
These functions are called the eigenfunctions, or characteristic functions, and the values
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are called the eigenvalues, or characteristic values, of the vibrating string. The set {λ1, λ2 …} is called the spectrum.
Discussion of Eigenfunctions We see that each represents a harmonic motion having the frequency λn /2π /2π = cn/2L cycles per unit time. This motion is called the nth normal mode of the string. The first normal mode is known as the fundamental mode and the others are known as overtones; musically they give the octave, octave plus fifth, etc. Since in (11)
the nth normal mode has n – 1 – 1 nodes, that is, points of the string that do not move (in addition to the fixed endpoints) as shown in figure given below.
Figure above shows the second normal mode for various values of t . At any instant the string has the form of a sine wave. When the left part of the string is moving down, the other half is moving up, and conversely. For the other modes the situation is similar. Tuning is done by changing the tension T . Our formula for the frequency λn /2π /2π = cn/2L of un with c =
√ confirms that effect
because it shows that the frequency is proportional to the tension.
Step 3. Solution of the Entire Problem. Fourier Series
The eigen functions (11) satisfy the wave equation (1) and the boundary conditions (2) (string fixed at the ends). A single un will generally not satisfy the initial conditions (3). But since the wave equation (1) is linear and homogeneous, it follows from Fundamental Theorem that the 6
sum of finitely many solutions un is a solution of (1). To obtain a solution that also satisfies the /L as before) initial conditions (3), we consider the infinite series (with λn = cn π …………….…………………….. (12)
Satisfying Initial Condition (3a) (Given Initial Displacement). From (12) and (3a) we obtain (0 ≤ x ≤ L).
…………….…………………….. (13)
f(x). Thus, by Hence we must choose the B n’s so that u(x,0) becomes the Fourier sine series of f(x). (4) in Sec. 11.3,
n = 1, 2 , …
…………….…………………….. (14)
Satisfying Initial Condition (3b) (Given Initial Velocity). Similarly, by differentiating (12) with respect to t and using (3b), we obtain
for t = 0 the derivative ∂u/∂t becomes the Fourier sine Hence we must choose the B*n’s so that for t series of g(x). Thus, again by (4) in Sec. 11.3,
Since λn = cn π /L, we obtain by division
n = 1,2, …
…………….…………………………………… (15)
x, t ) given by (12) with coefficients (14) and (15) is a Result. Our discussion shows that u ( x solution of (1) that satisfies all the conditions in (2) and (3), provided the series (12) converges 7
and so do the series obtained by differentiating (12) twice term wise with respect to x and t and 2 2 2 2 have the sums ∂ u/ ∂ x and ∂ u/ ∂t respectively, which are continuous. Solution (12) Established. According to our derivation, the solution (12) is at first a purely formal expression, but we shall now establish it. For the sake of simplicity we consider only the case when the initial velocity g(x) is identically zero. Then the B*n are zero, and (12) reduces to …………….…………………………………… (16)
It is possible to sum this series, that is, to write the result in a closed or finite form. For this purpose we use the formula
Consequently, we may write (16) in the form
These two series are those obtained by substituting x – ct and x + ct respectively, for the variable x in the Fourier sine series (13) for f(x). Thus …………….………………………………………………………… (17)
where f* is the odd periodic extension of f with the period 2 L as shown in figure below. Since the initial deflection f(x) is continuous on the interval 0 ≤ x ≤ L and zero at the endpoints, it follows x, t ) is a continuous function of both variables x and t for all values of the from (17) that u ( x variables. By differentiating (17) we see that u ( x x, t ) is a solution of (1), provided is twice differentiable on the interval 0 < x < L , and has one-sided second derivatives at x = 0 and x = L, x, t ) is established as a solution of (1), satisfying (2) which are zero. Under these conditions u ( x and (3) with g ( x x) = 0.
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f′(x) and f″(x) Generalized Solution. If f′(x) and f″(x) are merely piecewise continuous, or if those one sided derivatives are not zero, then for each t there will be finitely many values of x at which the second derivatives of u appearing in (1) do not exist. Except at these points the wave equation x, t ) as a “generalized solution,” as it is called, that will still be satisfied. We may then regard u ( x is, as a solution in a broader sense. For instance, a triangular initial deflection as in Example 1 (below) leads to a generalized solution. f* (x – ct ) is obtained from the graph Physical Interpretation of the Solution (17). The graph of f* of f*(x) f*(x) by shifting the latter ) units to the right as shown in figure given below. This means that f f *( x - ct )( x + ct ) *( x )(c > 0) represents a wave that is traveling to the right as t increases. Similarly, f *( *( x x, t ) is the superposition of these two waves. represents a wave that is traveling to the left, and u ( x
Example 1 Use Separation of Variables on the following partial differential equation.
Solution
In this case we’re looking at the heat equation with no sources and perfectly insulated boundaries. So, we’ll start off by again assuming that our product solution will have the form,
and because the differential equation itself hasn’t changed here we will get the same result from plugging this in as we did in the previous example so the two ordinary differential equations that we’ll need to solve are,
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Now, the point of this example was really to deal with the boundary conditions so let’s let’s plug the product solution into them to get,
Now, just as with the first example if we want to avoid the t rivial solution and so we can’t have for every t and so we must have,
Here is a summary of what we get by applying separation of variables to this problem.
Example 2 Use Separation of Variables on the following partial differential equation.
Solution
Now, as with the heat equation the two initial conditions are here only because they need to be here for the problem. We will not actually be doing anything with them here and as mentioned previously the product solution will rarely satisfy them. Again, the point of this example is only to get down to the two ordinary differential equations that separation of variables gives. So, let’s get going on that and plug the product solution, (we switched the G to an h here to avoid confusion with the g in the second initial condition) into the wave equation to get,
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Note that we moved the to the right side for the same reason we moved the k in the heat equation. It will make make solving the boundary value problem a little easier. Now that we’ve gotten the equation separated into a function of only t on the left and a function of only x on the right we can introduce a separation constant and again we’ll use so we can arrive at a boundary value problem that we are familiar with. So, after introducing the separation constant we get,
The two ordinary differential equations we get are then,
The boundary conditions in this example are identical to those from the first example and so plugging the product solution into the boundary conditions gives,
Applying separation of variables to this problem gives,
Example 3 Use
Separation
of
Variables
on
the
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following
partial
differential
equation.
Solution
Note that this is a heat equation with the source term of and is both linear and homogenous. Also note that for the first time we’ve mixed boundary condition types. At we’ve got a prescribed temperature and at x = L we’ve got a Newton’s law of cooling type boundary condition. We should not come away from the first few examples with the idea that the boundary conditions at both boundaries always the same type. Having them the same type just makes the boundary boundary value problem a little easier to solve in many cases. So we’ll start off with,
and plugging this into the partial differential equation gives,
Now, the next step is to divide by and notice that upon doing that the second term on the right will become a one and so can go on either side. Theoretically Theoreticall y there is no reason that the one can’t be on either side, however from a practical standpoint we again want to keep things a simple as possible so we’ll move it to the t side as this this will guarantee that we’ll get a differential equation for the boundary value problem that we’ve seen before. So, separating and introducing a separation constant gives,
The two ordinary differential equations that we get are then (with some rewriting),
Now let’s deal with the boundary conditions.
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and we can see that we’ll only get non-trivial non-trivial solution if,
So, here is what we get by applying separation of variables to this problem.
Applications of Partial Differential equations and separation of variables method Partial differential equations arise from the mathematical formulation of several physical and engineering problems in which the function (dependent variable) is a function of two or more independent variables, the most commonly occurring independent variables are those describing position and time. The partial differential equations have applications in electromagnetics with Maxwell’s equations, fluid dynamics, quantum mechanics, deformable solids, heat transfer, wave equations etc. The physical problems associated with partial differential equations may involve additional information arising from the physical situation. If this information is given on the boundary as boundary conditions, a boundary value problem results. If the information is given at one instant as initial conditions, an initial value problem results. It should be noted that the boundary conditions that are specified depend on the class of partial differential equations Thus, for an elliptic equation, the function or its derivatives will be specified around the entire boundary enclosing a region of interest, whereas for the hyperbolic and the parabolic equations, the functions cannot be specified around an entire boundary. Partial differential equations proved to be very important in solving many important engineering equations such as wave equations, heat transfer equation, Laplace equation etc. Given the almost universal applicability of numerical methods for the solution of partial differential equations, the question arises whether separation of variables with its severe restrictions on the type of equation and the geometry of the problem is still a viable tool and deserves further exposition. The answer is that the method of separation of variables still belongs to the core of applied mathematics. There are a number of reasons. Closed form (approximate) solutions show structure and exhibit explicitly the influence of the problem parameters on the solution. We think, for example, of the decomposition of wave motion into standing waves, of the relationship between driving frequency and resonance in sound waves, of the influence of diffusivity on the rate of decay of temperature in a heated bar, or of the generation of equipotential and stream lines for potential flow. Such structure and insight are not readily 13
obtained from purely numerical solutions of the underlying differential equation. Moreover, optimization, control, and inverse problems tend to be easier to solve when an analytic representation of the (approximate) solution is available. In addition, the method is not as limited in its applicability as one might infer from more elementary texts on separation of variables. Approximate solutions are readily computable for problems with time-dependent data, for diffusion with convection and wave motion with dissipation. Even domain restrictions can sometimes be overcome with embedding and domain decomposition techniques. Finally, there is the class of singularly perturbed and of higher dimensional problems where numerical methods are not easily applied while separation of variables still yields an analytic approximate solution. Equally important to us is the second reason for a new exposition of the method of separation of variables. We wish to emphasize the power of the method by solving a great variety of problems which often go well beyond the usual textbook examples. Many of the applications ask questions which are not as easily resolved with numerical methods as with analytic approximate solutions. Of course, evaluation of these approximate solutions usually relies on numerical methods to integrate, solve linear systems or nonlinear equations, and to find values of special functions, but these methods by now may be considered universally available "black boxes."
Conclusion All in all, Partial Differential are great tool in different fields of science and especially useful in the field of Engineering. The method of separation of variables is an important technique to solve these problems. In fact, what we see around us are the applications of mathematics. Using the knowledge of mathematics, we have invented many things and still we can bring many revolutions using this knowledge in a better manner.
References [1]. Kreyszig, E. (2011). Advanced engineering mathematics. (10th ed., pp. 545-550). NY: John Wiley and Sons, Inc. [2]. Cain G. & Meyer G.H. (2005). Separation of variables for partial differential equations: An eigenfunction approach. (2nd ed., pp. i-v). NY: CRC Press. [3]. Srivastava, A.C. & Srivastava, P.K. (2011). Engineering mathematics. (Eastern Economy ed., Vol. 2, p. 622). New Delhi: PHI Learning Private Limited. [4]. Dawkins,P. (2011). Separation of variables. Retrieved from http://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx [5]. Wojciechowski, S. R., & Marciniak, Marciniak, K. (2004).Separation of variables for differential equations. Department of Science and Technology, Liinkoping University, Linkoping, Sweden. Retrieved from http://webstaff.itn.liu.se/~krzma/publications/EncyclopediaMS282.pdf 14
