On the Solution of Reduced Wave Equation with Damping

In this paper we find particular solutions of Reduced wave equation with damping in the form ∆u+ k2n(x )u+ μ|∇u| = 0, Rn, μ ∈ R and n(x ) is a continuous function on Ω, by making use of Fundamental solution u= exp(ikR) R of the scalar Helmholtz equation and employing a variation of constant technique. Moreover, some examples are given to illustrate the importance of our results.


Introduction
Let x be an arbitrary point and y a fixed points in a domain 2 denote the distance between x and y.Let k > 0 be given such that R = nπ k , n = 1, 2, 3, . . .Consider the second-order partial differential equation of the form where µ is an arbitrary constant and n(x ) is a continuous function on Ω.
The reduced wave equation (or scalar Helmholtz equation) is an important partial differential equation that describes a variety of waves, such as sound, light and water waves.It arises in acoustic, electromagnetic and fluid dynamics for m = 2, 3. Helmholtz equation naturally appears from general conservation laws of physics and can be interpreted as a wave equation for monochromatic waves (wave equation in the frequency domain).Helmholtz equation can also be derived from the heat conduction equation, Schrödinger equation, telegraph and other wave type, or evolutionary, equations.In physically applications k and n(x ) are known wave number and refraction index of media respectively see details [5,8].
In [3,13] some particular solution have been constructed for scalar Helmholtz equation and reduced wave equation in two dimensions.Although 3-D problems are more realistic physically, their solutions are found rarely in the literature.Hence in this paper we investigate the solution of (1.1) for m = 3.
The general solution of the scalar Helmholtz equation with radial and polar coordinates in three dimensions can be obtained by using separation of variable as follows where j n (kr) and y n (kr) are the Bessel functions,and Y m l (θ , ϕ) is the spherical harmonics [1].
Very few elementary or closed-form solutions of the reduced wave equation are known for m = 3 for the case with variable index of refraction n.For layered media [n = n(x 2 )], only two solutions have been found so far.These are (i) Pekeris' solution [12] for a point source in a medium specified by n where i = −1 and R 2 = (x 1 − y 1 ) 2 +(x 2 + y 2 ) 2 +(x 3 − y 3 ) 2 ; and (ii) Kormilitsin's solution [10] for a line source extending parallel to the x 3 -axis in a media specified by n = x 2 , where In [9], R. L. Holford shows that an elementary solution of the reduced wave equation can be found for a line source extending parallel to the x 3 -axis in a medium specified by , where A, B, C, D and F are arbitrary constants, and for a point source excitation when n(x ) = A + C x 2 + F x 2 2 ; i.e., when the medium is layered.In both cases, the solution is obtained in the form where f and g are elementary functions of complex variable ζ, and C is a path running from ζ = 0 to ∞.
However in recent years a lot of useful numerical methods and numerical solutions have been presented for the reduced wave equation with variable coefficient (some times called nonlinear Helmholtz equation) [2,6,7].
Straightforward differentiation shows that for fixed satisfies Helmholtz equation, which is known as Fundamental solution of the Helmholtz equation [5,8].

Variation of Constant Method
Separating the fundamental solution of scalar Helmholtz equation into real and imaginary parts we have In this study our aim is to find particular solutions of (1.1) by using the above solutions of scalar Helmholtz equation.In particular, following the method of variation of constant we look for solutions of the form where f (R) is to be determined.We note that if we use u = 1 we do not obtain any new solution.Therefore, we use only (2.1).Naturally, we assume that f is a continuous function having first and second derivatives on Ω. Define where n(x ), µ are as defined above.
Lemma 2.1.Let f : Ω ⊂ R 3 → R\{0} be a continuous function that has first and second derivatives and satisfies the equation Proof.Using the chain rule we get for j = 1, 2, 3 Moreover, Where and Then we get and because of This completes the proof of lemma because of our assumption.
As an initial simplification, we will choose a constant k such that the (2.2) has no critical point.We can give an exponential type solution of (1.1) as follow.
Theorem 2.2.Let P and f satisfies (2.2) in Lemma 1 and φ be a continuously differentiable function, such that φ satisfies the Riccati differential equation ) is the solution of (1.1).
Proof.If we multiply the (2.2) by f 2 (R), then we get If we use the Riccati substitution where ψ is a continuously differentiable function, then we have If we substitute (2.8) in (2.6), we get (2.9) Again if we consider the change of variables where φ is a continuously differentiable function, then (2.9) becomes a classical Riccati equation in the form φ 4 , which completes the proof of the theorem.
where c is any constant (2.12) satisfies the partial differential equation It is known that the solution of (2.4) is equivalent to an exponential solution of the linear differential equation z = gz . (2.14) Here we note that,in order to construct the solution of (1.1) by using the Theorem1,we must solve the differential equation (2.4) or equivalently (2.14).But, in general to solve the differential equation (2.4) or (2.14) is not easy.Fortunately, we present an alternative solution procedure for differential equation (2.14) which works under the some condition on k 2 n(x ).

The Iteration Technique
Let w 0 = h 0 ∈ C ∞ (a, b) and consider the equation For some w 0 function we shall give a new method to obtain the particular solutions of equation (2.14).This method depends on finding some symmetric structure as in [4] by using asymptotic behavior of equation (2.14).
Thus for this purpose if we differentiate (3.1) with respect to the t, we find that Again if we differentiate (3.2) with respect to the t, we find that where w 1 = w 0 + h 0 and h 1 = h 0 +w 0 w 0 .
If we differentiate again (3.3) with respect to the t, we find that where Thus if we continue in this way,we get for n ≥ 0, and similarly where From the ratio of the (n + 4)th and (n + 3)th derivatives, we get If we have, for sufficiently large n ≥ 0, w n which yields But in Eq. (2.10) the integrand function is Then (3.10) becomes Substituting (3.11) into (3.5)we obtain the first-order differential equation Thus we get the general solution of (3.12) as where c 1 and c 2 are arbitrary constants.Note that in [11] a different method with same procedure has been applied and same result has been obtained for µ = 0.
Example 3.4.Let k > 0, µ and n(x be given, then g = −12R 3−R 3   and γ = Thus if we use the same procedure of Example 1, we get the solution of the partial differential equation (1.1) of the form + (23) where c 1 and c 2 are arbitrary constants.is the solution of (1.1).
Proof.If there exists a γ, which satisfies (2.9), then a solution of (1.1) is where c 1 and c 2 are arbitrary constants.
Thus the solution of the given differential equation is . Remark 3.8.In order to calculate γ in Examples 3.1, 3.2, 3.3 and 3.4 MATHEMATICA software has been used.Conclusion 3.9.(1.1) has many application in physics, chemistry and some branch of engineering.Thus the solution of (1.1) is important in these areas in the applications.It is shown that if the solution of (1.1) is in the form (2.1) then f must be solved from (2.2).In Section 3, a functional iteration method of a linear equation of the form L(z) = z − λ 0 z = 0 is given.If during the iteration process (3.9) is obtained at some step, then Theorem 1 gives the particular solutions of (1.1).

Corollary 2 . 3 .
If there exists a continuous function P such that k 2 n(x ) = k 2 − P and f satisfies the differential equation P

3 = − 1+4R 4 R+R 5 .
If we substitute this value of γ in (3.13) we get the solution of the partial differential equation (1.1) of the form

Conclusion 3 . 10 .
The boundary value problem which include Helmholtz equation or reduced wave equation (with damping term) with Dirichlet or Neumann condition in any domain can represent a physical problem.Some of them can be given as acoustic scattering, inverse acoustic scattering or inverse conductive scattering problem in any homogeneous or inhomogeneous media, ocean waves problem and ext.The arbitrary constants, which are obtained by the iteration method of general solutions have special importance for each boundary value problems which are mentioned above.