Laplace transform pairs of N-dimensions and second order linear partial differential equations with constant coefficients

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1 Annales Mathematicae et Informaticae 35 8) pp. 3 Laplace transform pairs of N-dimensions and second order linear partial differential equations with constant coefficients A. Aghili, B. Salkhordeh Moghaddam Department of Mathematics, Faculty of Science University of Guilan, Rasht, Iran Submitted 5 February 8; Accepted 5 September 8 Abstract In this paper, authors will present a new theorem and corollary on multidimensional Laplace transformations. They also develop some applications based on this results. The two-dimensional Laplace transformation is useful in the solution of partial differential equations. Some illustrative examples related to Laguerre polynomials are also provided. Keywords: Two-dimensional Laplace transforms, second-order linear nonhomogenous partial differential equations, Laguerre polynomials. MSC: 44A3, 35L5. Introduction In [3 R. S. Dahiya established several new theorems for calculating Laplace transform pairs of N-dimensions and two homogenous boundary value problems related to heat equations were solved. In [4 J. Saberi Najafi and R. S. Dahiya established several new theorems for calculating Laplace transforms of n-dimensions and in the second part application of those theorems to a number of commonly used special functions was considered, and finally, by using two dimensional Laplace transform, one-dimensional wave equation involving special functions was solved. Later in [, authors, established new theorems and corollaries involving systems of two-dimensional Laplace transforms containing several equations. The generalization of the well-known Laplace transform L[ft); s 3 e st ft)dt,

2 4 A. Aghili, B. Salkhordeh Moghaddam to n-dimensional is given by L n [f t ); s exp s t )f t )P n d t ), where t t, t,..., t n ), s s, s,...,s n ), s t n i s it i and P n d t) n k dt k. In addition to the notations introduced above, we will use the following throughout this paper. Let t υ t υ, tυ,..., tυ n ) for any real exponent υ and let P k t ) be the k-th symmetric polynomial in the components t k of t. Then P t υ ), P t υ ) t υ + tυ tυ n, P t υ ) n i,j,i<j tυ i tυ j,. P n t υ ) t υ tυ...tυ n. The inverse Laplace transform is given by ) n a+i L [F s); t iπ. The main theorem Theorem.. Let d+i d i c+i c i e s t F s)p n s)d s. gs) L[ft); s, Fs) L[t 3/ g/t); s, Hs) L[tft 4 ); s. If ft), t 3/ g t ) and tft4 ) are continuous and integrable on, ), then where n,,..., N. L n [P n t / )F P t ) ) ; s 4π n+)/ H[ P s / ), P n s / ) Proof. We have g ) t exp u ) fu)du..) t Multiply both sides of.) by t 3/ exp st), Res) > and integrate with respect to t on, ) to get e st g t ) dt t 3/ e st e u t f u)t 3/ dudt..)

3 Laplace transform pairs of N-dimensions... 5 Since the integral on the right side of.) is absolutely convergent, we may change the order of integration to obtain e st g t ) dt t 3/ f u) Evaluating the inner integral on the right side of.3), we get F s) f u)e su π du. u e st u/t t 3/ dt du..3) Now, on setting u v 4, replacing s by P t ) and then multiplying both sides of.3) by P n t / )e s t and integrating with respect to t, t,..., t n from to, leads to the statement. Corollary.. Letting n we get from Theorem., that L F xy x + ) ) } ; u, v 4π 3/ H[ u + v)..4) y uv As an application of the above theorem and corollary, some illustrative examples in two dimensions are also provided. Example.3. Let ft) sin t), then Fs) π Hs) + π 8 s cos s 4 +4s, ) ) ) s S +s sin π s 4 ) ))} s C, π hence [ xy) 3 L 4x + y) + x y, u, v π uv π u + v)cos u + v) ) S + u + v)sin u + v) ) C ) ) u + v) π )) u + v) + } π, π where Fresnel s integrals are defined as following Cx) π x Example.4. If ft) lnαt) then cost) t dt, Sx) π x sint) t dt. Fs) lnα/s) γ} s and Hs) lnα) + 4 γ lns))}. s

4 6 A. Aghili, B. Salkhordeh Moghaddam Using.4), we arrive at [ ) ) xy 4x + y) L ln x + y αxy) γ, u, v π 4 ln u + v) lnα) + 6 ln) + 4γ ) uv u + v). In the following example, we give an application of two-dimensional Laplace transforms and complex inversion formula for calculating some of the series related to Laguerre polynomials. Example.5. We shall show that see [6). n L nx)l n y)λ n λ λx+y) e λ λxy I λ. n L nt)l n ξ) e t δt ξ), where L n x) is Laguerre polynomial and I x) is modified Bessel s function of order zero. Solution. n.. It is well known that L[L n x), p p p) Taking two-dimensional Laplace transform of the left hand side, leads to the following [ ) L L n x)l n y)λ n, p, q L n x)l n y)λ n e px qy dxdy. n Changing the order of summation and double integration to get [ L L n x)l n y)λ n, p, q L n x)l n y)λ n e px qy dxdy. n The value of the inner integral is λ n n n λ λ n pq n n L n x)l n y)e px qy dxdy ) n ) n } p q ), λ pq + kp + q) k, where k λ. Using complex inversion formula for two-dimensional Laplace transform to obtain, L n x)l n y)λ n n ) a+i iπ d+i d i e px+qy dp dq λ pq + kp + q) k

5 Laplace transform pairs of N-dimensions... 7 λ iπ λ iπ a+i a+i iπ d+i d i } e px pq + kp + q) k dp e qy dq e kxq ) q+k q + k eqy dq λx+y) λ e λ ) λxy I. λ. Taking two-dimensional Laplace transform of the left hand side, leads to the following [ ) L L n t)l n ξ), p, q L n t)l n ξ)e pt qξ dt dξ. n Changing the order of summation and double integration to get, [ L L n t)l n ξ), p, q L n t)l n ξ)e pt qξ dt dξ. n It is not difficult to show that the value of the inner integral is L n t)l n ξ)e pt qξ dt dξ ) n pq p q and n n n ) n n pq p q) p + q. Using complex inversion formula for two-dimensional Laplace transforms to obtain, L n t)l n ξ) n iπ ) a+i b+i b i ) n e pt+qξ dp dq. p + q The above double integral may be re-written as follows, L n t)l n ξ) a+i } e qξ b+i e pt πi πi p q) dp dq. n b i The value of the inner integral by residue theorem is equal to e q)t, upon substitution of this value in double integral we get, n L n t)l n ξ) πi a+i e qξ e q)t dq e t πi a+i e qt ξ) dq, therefore L n t)l n ξ) e t δt ξ). n

6 8 A. Aghili, B. Salkhordeh Moghaddam 3. Solution to second-order linear partial differential equations with constant coefficients The general form of second-order linear partial differential equation in two independent variables is given by see [5). Au xx + Bu xy + Cu yy + Du x + Eu y + Fu qx, y), < x, y <, 3.) where A, B, C, D, E and F are given constant and qx, y) is source function of x and y or constant. We will use the following for the rest of this section see [5, 6). If ux, ) fx), u, y) gy), u y x, ) f x), u x, y) g y), u, ) u 3.) and if their one-dimensional Laplace transformations are Fu), Gv), F u) and G v), respectively, then L [ux, y); u, v ux, t)e ux vt dxdt Uu, v), L [u xx ; u, v u Uu, v) ugv) G v), L [u xy ; u, v uvuu, v) ufu) vgv) u, ), 3.3) L [u yy ; u, v v Uu, v) ufu) F u), L [u x ; u, v uuu, v) Gv), L [u y ; u, v vuu, v) Fu). Applying double Laplace transformation term wise to partial differential equations and the initial-boundary conditions in 3.) and using 3.3), we obtain the transformed problem Uu, v) Au + Cv + Buv + Ev + Du + F AuGv) + G v)) + BuFu) + vgv) u ) + CvFu) + F u)) + DGv) + EFu) + Qu, v)}. 3.4) Now, in the following examples we illustrate the above method. Example 3.. Letting A B C, we get Du x + Eu y + Fu qx, y), < x, y <, E/D > ). With initial boundary conditions ux, ) fx), u, y) gy),

7 Laplace transform pairs of N-dimensions... 9 application of the relationship 3.4) gives Uu, v) DGv) + EFu) + Qu, v). 3.5) Ev + Du + F The inverse double Laplace transform of 3.5) leads to the formal solution ux, y) e F D x g y ED ) x + e F E y f x DE ) y x + D e F D ξ qx ξ, y E D ξ)dξ, if y > E D x, y e F E η qx D E η, y η)dη, if y < E D x. E Example 3.. If C E D, A, B α, F β, then 3.) reduces to With the following initial conditions we obtain Uu, v) u xx + αu xy + βu qx, y), < x, y <. u, y) gy), u x, y) g y), ux, ), u, ) u u + αuv + β ugv) + G v) + αvgv) u ) + Qu, v)}. 3.6) The inverse double Laplace transform of 3.6) yields see [7) [ [ ux, y) L Qu, v) [Uu, v) L u + L ugv) + αuv + β u + αuv + β [ [ [ + L G v) u + αl vgv) + αuv + β u + αu L + αuv + β u + αuv + β or equivalently ux, y) x ξ J ) βηx ξ) qξ η, y αη)dη dξ + gy αx) + αx ) βη βη α αx η J α x η α ) gy η)dη + αx ) βη J α α x η α ) g y η)dη + gy) gy αx) + αx βη αx ) ) βη J α αx η η α x η α ) gy η)dη +, if y > αx, αu J α βyαx η)), if y < αx.

8 A. Aghili, B. Salkhordeh Moghaddam 4. Conclusions The multi-dimensional Laplace transform provides powerful method for analyzing linear systems. It is heavily used in solving differential and integral equations. The main purpose of this work is to develop a method of computing Laplace transform pairs of N-dimensions from known one-dimensional Laplace transform and making continuous effort in expanding the transform tables and in designing algorithms for generating new inverses and direct transform from known ones. It is clear that the theorems of that type described here can be further generated for other type of functions and relations. These relations can be used to calculate new Laplace transform pairs. Acknowledgements. The authors would like to thank referees for their comments and questions. References [ Aghili, A., Salkhordeh Moghaddam, B., Laplace transform pairs of n-dimensions and a Wave equation, Intern. Math. Journal, 54) 4) [ Aghili, A., Salkhordeh Moghaddam, B., Multi-dimensional laplace transform and systems of partial differential equations, Intern. Math. Journal., 6) 4. [3 Dahiya, R.S., Vinayagamoorty, M., Laplace transform pairs of N-dimensions and heat conduction problem, Math. Comput. Modelling., 3) 99) [4 Dahiya, R.S., Saberi-Nadjafi, J., Theorems on N-dimensional laplace transforms and their applications, 5th annual Conference of Applied Mathematics, Univ. of Central Oklahoma, Electronic Journal of Differential Equations, 999) [5 Ditkin, V.A., Prudnikov, A.P., Operational calculus in two variables and its application, New York, 96). [6 Roberts, G.E., Kaufman, H., Table of taplace transforms, Philadelphia, W. B. Saunders Co., 966). A. Aghili B. Salkhordeh Moghaddam Department of Mathematics Faculty of Sciences Namjoo St., Rasht Iran armanaghili@yahoo.com salkhorde@yahoo.com