Distribution Solutions of Some PDEs Related to the Wave Equation and the Diamond Operator

Transcription

1 Applied Mathematical Sciences, Vol. 7, 013, no. 111, HIKARI Ltd, Distribution Solutions of Some PDEs Related to the Wave Equation and the Diamond Operator Chalermpon Bunpog Department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 5000, Thailand Copyright c 013 Chalermpon Bunpog. 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. Abstract In this paper, we find a distribution solution of some PDEs of the operator k, defined by (1), which is related to the wave equation and the diamond operator. The existence of the solution to these equations are proven. Mathematics Subject Classification:46F10 Keywords: Distribution, Wave equation, Diamond operator. 1 Introduction Many problems in differential equations involve with a derivative of the Dirac delta distribution, for examples, see([], [8], [9], [1], [13]). In this work, first of all, we find a distribution solution of a δ-derivative equation of the partial differential operator k, of the form k y(x) = a r r δ(x),

2 5516 Chalermpon Bunpog where m is a nonnegative number and a r is a constant with a 0 0. The operator k is defined by ( p k = x a=1 a ) 4 ( n x b=p+1 b ) 4 k. (1) Some properties of the operator k are studied in [3], [4] and [5]. In fact, the operator k can be expressed as the product of the operators k and k, that is k = k k, where these operators are defined by ( p k = x a=1 a ) ( n x b=p+1 b ) k, () and ( p k = x a=1 a ) ( n + x b=p+1 b ) k. (3) The famous operator k is known as the diamond operator, first introduced by A. Kananthai []. It can be expressed as the product of the Laplace operator iterated k-times k and the ultra-hyperbolic operator iterated k-times k, that is k = k k. Therefor, the operator k can be expressed as k = k k k. Second of all, we use some results from the study of a distribution solution of a δ-derivative equation of the partial differential operator k to find a general distribution solution of the nonhomogeneous equation k y(x) =f(x). Finally, we modify the operator k by k 1,k,k 3 = k 1 k k 3 to obtain the general distribution solution of the wave equation and the general distribution solution of the nonhomogeneous equation of the diamond operator. This paper is organized into 3 sections. In section, the preliminary note are given. It includes necessary Definitions and Lemmas which will be used in the proof of the main results. In section 3, the main results are presented and proven. Preliminary Notes Definition.1 Let x =(x 1,x,...,x n ) R n, v = x 1 + x + + x n and Γ be the Gamma function. For any complex number α, we define the function

3 Distribution solutions of some PDEs 5517 Rα e (v) by ( ) α n n α Rα e (v) =Γ v α π ( n ) Γ( α ). (4) The function Rα e (v) is called the elliptic kernel of Marcel Riesz. It is an ordinary function if Re(α) n and it is a distribution of α if Re(α) <n. Definition. Let x =(x 1,x,...,x n ) R n and u = x 1 + x + + x p x p+1 x p+ x n be a nondegenerated quadratic form. Denote the interior of the forward cone by Ϝ = {x R n x 1 > 0,u > 0}. The function Rβ H (u) is defined by { Rβ H (u) u β n K n(β), for x Ϝ; (5) 0, for x Ϝ, where K n (β) = π n 1 Γ ( ) ( +β n Γ 1 β ) Γ(β) Γ ( ) ( +β p Γ p β ), and β is a complex number. The function Rβ H (u) is called the ultra-hyperbolic kernel of Marcel Riesz. Note that Rβ H (u) is an ordinary function if Re(β) n and it is a distribution of β if Re(β) <n. Definition.3 Let x =(x 1,x,...,x n ) C n,w= x 1 + x + + x p i(x p+1 + x p+ + + x n) and z = x 1 + x + + x p + i(x p+1 + x p+ + + x n), where i = 1. We define functions S γ (w) and S γ (w) by S γ (w) = Γ ( ) n γ γ n w γ π ( n ) Γ( γ ), (6) for any complex numbers γ and η. T η (z) = Γ ( ) n η η n z η π ( n ) Γ( η ), (7) Lemma.4 Let D be a differentiation operator, δ(x) be the Dirac delta distribution and y(x) be any distribution, where x R n. Then Proof. See([11], p ). Dy(x) =Dδ(x) y(x) =Dy(x) δ(x).

4 5518 Chalermpon Bunpog Lemma.5 Consider the equation k y(x) =0for x R n and k is the diamond operator iterated k-times, defined by (). We have y(x) =( 1) k Rk e ( (v) R H k (u) ) ( n 4 ) is the solution of such the equation, where R e k (v) is defined by (4) with α =k and Rk H (u) is defined by (5) with β =k. ( n 4 ) denote the derivative n 4 times for an even number n 4. Proof. See([1], p. 1379). Lemma.6 Let k be define by (3), consider the equation k y(x) =δ(x) for x R n. We have y(x) = S k (w) T k (z) is the solution of such the equation, where S k (w) and T k (z) are define by (6) and (7) respectively, with γ = η =k. Proof. See([5], p. 3). Lemma.7 For k N we define G k (x) =( 1) k R e k(v) R H k(u) S k (w) T k (z), where R e k (v), RH k (u), S k(w) and T k (z) are defined by (4), (5), (6) and (7), respectively, with α = β = γ = η =k. Then G k (x) = v R e(v) u R H(u) w S (w) z T (z), k =, 3,.... 4() ((k 1)!) 4 (n (l + 1)) 4 Moreover, G k (x) is a tempered distribution. Proof. First, we will show that (8) R e k (v) = v R(v) e. (9) (k 1)! (n (l + 1)) From (4), by substituting α =4, we have R e 4 (v) =Γ ( n 4 ) v 4 n 4 π ( n ) Γ().

5 Distribution solutions of some PDEs 5519 Since Γ(z + 1)= zγ(z), R4(v) e =R()(v) e v = (n 4) Γ ( n ) n v π ( n ) Γ(1) v = (n 4) Re (v), and R6(v) e =R(3)(v) e = Γ ( ) 6 n n 6 v 6 π ( n ) Γ(3) v =!(n 4)(n 6) Re (v). By induction on k, we obtain Similarly, and R e k (v) = S k (w) = T k (z) = Next, we want to show that v R e(v). (k 1)! (n (l + 1)) w S (w). (10) (k 1)! (n (l + 1)) z T (z). (11) (k 1)! (n (l + 1)) Rk H (u) = ( 1) u R H(u). (1) (k 1)! (n (l + 1)) From ([7] Lemma.9(Eq.13)), by substituting ν = 0 and replacing k by k 1, we have Rk(u) H u R H = (u). (k 1)! ( + l n)

6 550 Chalermpon Bunpog Furthermore, since (+l n) = (n (l+1)), (1) is true. By using (9), (10), (11), (1) and properties of convolution, we obtain (8) immediately. Finally, we shall show that G k (x) is a tempered distribution. Choose supp w S (w) =A, where A is a compact subset of Ϝ. Then by ([13], p.15), the convolution w S (w) z T (z) is a tempered distribution and by ([10], p.153), the convolution Rα(v) e Rβ H(u) is also a tempered distribution. Hence G k(x) isa tempered distribution. The proof is completed. Lemma.8 Consider the equation k y(x) =δ(x) for x R n and k is defined by (1). We have y(x) =G k (x) is the solution of such the equation, where G k (x) is defined by Lemma.7. Proof. See([5], p. 6). Lemma.9 Let k be the operator defined by (1) and G k (x) is defined as in Lemma.7. We have that (1) If 0 m<k, then m G k (x) =G k m (x) and () If m k, then m G k (x) =G 1 k m (x), where x C supp G k m for a compact subset C of R n such that C supp G k m. Proof. (1) Let 0 m<k, from Lemma (.8) we have k m m G k (x) =δ(x). We take a convolution both sides by G k m (x) and use Lemma (.4). We obtain k m G k m (x) m G k (x) =G k m (x). Hence m G k (x) =G k m (x), 0 m<k. () For m k, we have m G k (x) = m k δ(x) k G k (x) = m k δ(x). Let y(x) = m k δ(x). Take a convolution both sides by G m k (x). We obtain G m k (x) y(x) = δ(x). Let C be a compact subset of R n such that C supp G m k. Thus for x C supp G m k, the function G m k lies in a set of a subalgebra convolution. And since G m k is nonzero, by ([6], p ), we have that the equation G m k (x) y(x) =δ(x) has the unique solution y(x) = G 1 m k (x), where G 1 m k is the inverse convolution algebra of G m k. Therefor, m G k (x) =G 1 k m (x), x C supp G m k for m k. Then the proof is completed.

7 Distribution solutions of some PDEs Main Results In this section, we describe a solution of some equations of the operator k which is relate to the wave equation and the diamond operator. We use the results in previous section to show the existence of a solution. Theorem 3.1 Consider the equation k y(x) = a r r δ(x), (13) where k is defined by (1), y(x) is an unknown function, δ(x) is The Dirac delta distribution, m is a nonnegative number and a r is a constant with a 0 0. Then y(x) = a r G k r (x)+ r=k a r G 1 r k (x), x m r=k supp G r k C, for a compact subset C of R n such that m r=k supp G r k C, where G k is defined by Lemma (.7) and G 1 k is the inverse convolution algebra of G k. Proof. We take a convolution both sides of (13) by G k (x). Use Lemma (.4), Lemma (.8) and properties of a convolution. We obtain that y(x) = = a r r G k (x), a r r G k (x)+ a r r G k (x). From equation (8), we have m r=k supp G r k. Choose a compact subset C of R n such that m r=k supp G r k C. Thus by Lemma (.9), we obtain that y(x) = a r G k r (x)+ r=k r=k a r G 1 r k (x), for x m r=k supp G r k C. The proof is completed. Theorem 3. Consider the equation k y(x) =f(x), (14) where k is defined by (1), y(x) is an unknown function and f(x) is a given distribution. Then the general solution of (14) is y(x) =( 1) k R e k(v) ( R H k (u) ) ( n 4 ) Sk (w) T k (z)+g k (x) f(x), (15)

8 55 Chalermpon Bunpog where functions Rk e (v), RH k (u), S k(w), T k (z), and G k (x) are defined by (4), (5), (6), (7), and in Lemma (.7) respectively. ( ) n 4 denote the derivative n 4 times for an even number n 4. Proof. First, we find a solution of the homogeneous equation k y(x) =0. Since k = k k, that is k k y(x) = 0. From this equation and Lemma (.5), we have k y(x) =( 1) k R e k (v) ( R H k (u)) ( n 4 ). Take a convolution both sides of this equation by S k (w) T k (z) and use Lemma (.4) and Lemma (.6), we obtain that y(x) =( 1) k R e k(v) ( R H k (u) ) ( n 4 ) Sk (w) T k (z) (16) is the solution of the homogeneous k y(x) = 0. Next, we want to find a particular solution of (14). We take a convolution by G k (x) to both sides of (14), by Lemma (.4) and Theorem (3.1) with m = 0 and a 0 = 1, then y(x) =G k (x) f(x) (17) is the particular solution of (14). Hence, the general solution (15) of (14) is obtained by (16) and (17). We have completed the proof. In addition, we modify the operator k by k 1,k,k 3 = k 1 k k 3 where k 1,k and k 3 is nonnegative integers. Let k 1 = k 3 =0,k = k, p =1,x 1 = t and replacing n by n+1, the operator k 1,k,k 3 is reduced to the n dimensional wave operator iterated k-times ( ) k k = t. x x n+1 In this case, the equation (14) is reduced to the wave equation From (15), we obtain that k y(x) =f(x). (18) y(x) = ( R H k (u)) ( n 3 ) + R H k (u) f(x), is the general distribution solution of the n-dimensional wave equation (18), where n is odd number and n 3. Let k 1 = k = k and k 3 = 0, the operator k 1,k,k 3 is reduced to the diamond operator iterated k times k. In this case the equation (14) is reduced to k y(x) =f(x).

9 Distribution solutions of some PDEs 553 From (15), we obtain that y(x) =( 1) k Rk e (v) ( Rk H (u)) ( n 4 ) +( 1) k Rk e (v) RH k (u) f(x), is the general distribution solution of the the diamond operator iterated k times. ACKNOWLEDGEMENTS. This research was supported by Chiang Mai University. References [1] A. Kananthai, On the diamond operator related to the wave equation, Nonlinear Analysis, 47(001) p [] A. Kananthai, On the solution of the n-dimensional diamond operator, Applied Mathematics and Computation, 88(1997), [3] A. Kananthai and S. Suantai, On the convolution product of the distributional kernel K α,β,γ,ν, International Journal of Mathematics and Mathematical Sciences, 3(003), [4] A. Kananthai and S. Suantai, On the inversion of the kernel K α,β,γ,ν, related to the operator k, Indian J. pure appl. Math., 34(003), [5] A. Kananthai, S. Suantai and V. Longani, On the operator k related to the wave equation and Laplacian, Applied Mathematics and Computation, 13(00), [6] A. H. Zemanian, Distribution Theory and Transform Analysis, Mc-Graw Hill, New York, (1964). [7] C. Bunpog, The compound equation related to the Bessel-Helmholtz equation and the Bessel Klein-Gordon equation, Applied Mathematical Sciences, 7(013), p [8] D. S. Jones, The theory of generalised functions, McGraw-Hill, (1966). [9] I. M. Gel sfand and G. E. Shilov, Generalized functions, Academic Press, Inc., (1964). [10] M. A. Tellez, The distribution Hankel transform of Marcel Riesz s ultrahyperbolic kernel, Studies in Applied Mathematics 93(1994),

10 554 Chalermpon Bunpog [11] R. F. Hoskins and J. S. Pinto, Distributions, ultradistributions and other generalized functions, Ellis Horwood Limited, [1] S. E. Trione, On the Solutions of the Causal And Anticausal n - Dimensional Diamond Operator, Integral Transforms and Special Functions, 13(00), p [13] W. F. Donoghue, Distributions and Fourier transform, Academic Press, Inc., (1969). Received: August 10, 013