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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 761248, 19 pages
Spectral Approach to Derive the Representation Formulae for Solutions of the Wave Equation
Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
Received 28 September 2011; Accepted 30 January 2012
Academic Editor: Pablo González-Vera
Copyright © 2012 Gusein Sh. Guseinov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Using spectral properties of the Laplace operator and some structural formula for rapidly decreasing functions of the Laplace operator, we offer a novel method to derive explicit formulae for solutions to the Cauchy problem for classical wave equation in arbitrary dimensions. Among them are the well-known d'Alembert, Poisson, and Kirchhoff representation formulae in low space dimensions.
The wave equation for a function of space variables and the time is given by where is the Laplacian. The wave equation is encountered often in applications. For the equation can represent sound waves in pipes or vibrations of strings, for waves on the surface of water, for waves in acoustics or optics. Therefore, formulae that give the solution of the Cauchy problem in explicit form are of great significance. In the Cauchy problem (initial value problem) one asks for a solution of (1.1) defined for , that satisfies (1.1) for , and the initial conditions
Passing to an arbitrary let us denote by the solution of the problem It is easy to see that then the function is the solution of the problem Indeed, integrating (1.7) we get Hence, by the second condition in (1.8). On the other hand, from (1.9), Comparing (1.13) and (1.14), we get (1.10). Besides, so that initial conditions in (1.11) are also satisfied.
In the present paper, we give a new proof of formulae (1.17), (1.18) for the solution of problem (1.7), (1.8). Our method of the proof is based on the spectral theory of the Laplace operator. We hope that such a method may be useful also in some other cases of the equation and space.
The paper consists, besides this introductory section, of three sections. In Section 2, we describe the structure of arbitrary rapidly decreasing function of the Laplace operator, showing that it is an integral operator and giving an explicit formula for its kernel. Next we use these results in Section 3 to derive the explicit representation formulae for the classical solution to the initial value problem for the wave equation in arbitrary dimensions. The final Section is an appendix and contains some explanation of several points in the paper.
2. Structure of Arbitrary Function of the Laplace Operator
Let be the self-adjoint positive operator obtained as the closure of the symmetric operator determined in the Hilbert space by the differential expression on the domain of definition that is the set of all infinitely differentiable functions on with compact support. Let denote the resolution of the identity (the spectral projection) for A: Next, let be any infinitely differentiable even function on the axis with compact support and its Fourier transform. Note that the function tends to zero as faster than any negative power of . Consider the operator defined according to the general theory of self-adjoint operators (see ):
The following theorem describes the structure of the operator showing that it is an integral operator and giving an explicit formula for its kernel in terms of the function .
Theorem 2.1. The operator is an integral operator Further, there is a smooth function defined on the interval such that The function depends on the function as follows. If one sets then where denotes the th order derivative of . Further, if , then . For any solution of the equation the equality holds for .
Proof. First we consider the case . In this case, the statements of the theorem take the following form: for ; the operator is an integral operator of the form
and for any solution of the equation
To prove the last statements note that, in the case , the operator is generated in the Hilbert space by the operation and the operator by the operation . The resolvent of the operator has the form while the spectral projection of the operator has the form (see [3, page 201]) Therefore, where we have used the inversion formula for the Fourier cosine transform. Therefore, (2.11) is proved. To prove (2.13) note that the general solution of (2.12) is where and are arbitrary constants. Then, we have, for , where we have used the fact that the function is even and therefore The same result can be obtained similarly for . Thus, (2.13) is also proved.
Now we consider the case . We shall use the integral representation of the resolvent of the operator . As is known [4, Section 13.7, Formula (13.7.2)], where is the Hankel function of the first kind of order . Next, according to the general spectral theory of self-adjoint operators [3, page 150, Formula (11)], we have Therefore, from (2.4) it follows that the representation (2.5) holds with Now the representation (2.6), which expresses that is a function of , follows from (2.23) by (2.21).
To prove (2.10) we use (2.23). By virtue of (2.23), see Appendix. Here we have used the fact that from (2.9) it follows that that is, and therefore
Finally, to deduce the explicit formulae (2.7), (2.8), we take in (2.10). Then, putting , we can write If we set then the left-hand side of (2.28) equals On the other hand, where is the surface area of the -dimensional unit sphere ( is the gamma function) and denotes the surface element of the sphere . Therefore, setting we get that (2.28) takes the form Substituting here the expression of given in (2.29) and making then the change of variables , we obtain Hence (2.7) follows. Further, it is not difficult to check that the formula (2.33) for is equivalent to (2.8), see Appendix.
Since is smooth and has a compact support, it follows from (2.7), (2.8) that the function also is smooth and has a compact support; more precisely, if , then . This implies, in particular, convergence of the integral in (2.10) for each fixed . The theorem is proved.
For , , let us set Since applying (2.9), (2.10), we get Hence, by the inverse Fourier transform formula, Multiplying both sides of the last equality by and then integrating on , we get Substituting here for its expression and setting we obtain Obviously, the function defined by (3.9) is the solution of problem (3.1), (3.2). Next we will transform the left-hand side of (3.10) using Theorem 2.1.
First we consider the case . In this case, (3.10) takes the form and from (2.7), (2.8) we have Therefore, making the change of variables and taking into account the evenness of the function , we can write Substituting this in the left-hand side of (3.11), we obtain Hence, by the arbitrariness of the smooth even function with compact support, we get
Further assume that . Making the change of variables where is the surface element of the unit sphere , we get Further, making in the right-hand side of (3.17) the change of variables where is the surface element of the sphere , we have Therefore, and (3.10) becomes Consider the cases of odd and even separately.
Let . Then, by (2.8) we have and it follows from (2.7) (by successive differentiation) that Therefore, and (3.21) takes the form Further, integrating times by parts, we get where Since is identically zero for large values of , we have from (3.27) that . Also, it follows directly from (3.27) that . Therefore, (3.25) becomes Since in (3.28) is arbitrary smooth even function with compact support, we obtain that This coincides with (1.17) by (3.19).
Now let us consider the case . In this case, by (2.8) we have and therefore Hence, setting we get Substituting this in the left-hand side of (3.21) (beforehand replacing by in the left side of (3.21)), we obtain or, using (3.23), Further, integrating times by parts, we get where Since is identically zero for large values of , we have from (3.37) that . Also, using the expression of , we can check directly from (3.37) that . Therefore, (3.35) becomes Since in (3.39) is arbitrary smooth even function with compact support, we obtain that This coincides with (1.18) by (3.32).
AppendixFor reader’s convenience, in this section we give some explanation of several points in the paper.(1)Let us show how (2.33) for implies (2.8).
In the case with , (2.33) takes the form Hence, Therefore, taking into account that by virtue of we have we get In the right-hand side we replace by , then divide both sides by and integrate on to get because for any , using the change of variables , we have Therefore, differentiating (A.10) with respect to , we get Thus, (2.8) is obtained also for with .
(2) Here we explain (2.24). Note that since the spectrum of the operator is (zero is included into the spectrum), the spectral representation formula (2.4) should be understood in the sense of the formula where is an arbitrary positive real number and the integral does not depend on ( is zero on because is a positive operator). Therefore, for (2.24) we have to show that for any .
Since for any we have Given , we can choose a such that since the function is continuous for (we choose the continuous branch of the square root for which ). Further, we choose a number such that for sufficiently large positive number . This is possible by (2.3) and the fact that has a compact support. Let us set . Then, Further,
(5) The explicit formula for the solution of the wave equation in the case even can be derived from the case odd by a known computation called the “method of descent” (see ).
(6) Since for , , we have , and on the left-hand side of (2.10) the integral is taken in fact over the ball , for fixed . Therefore, this integral is finite for each and any solution of (2.9). We proved (2.10) for . If the solution is an analytic function of , then (2.10) will be held also for complex values of by the uniqueness of analytic continuation.
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