International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 41 Note on Triple Aboodh Transform and Its Application S. Alfaqeih, T. ÖZIS Faculty of Science Department of Mathematics, 35100 Ege University, Bornova – Izmir – TURKEY sfaqeih@yahoo.com, turgut.ozis@ege.edu.tr Abstract- In this paper, we introduce the definition of triple Aboodh transform, some properties for the transform are presented. Furthermore, several theorems dealing with the properties of the triple Aboodh transform are proved. In addition, we use this transform to solve partial differential equations with integer and non-integer orders. Keywords-triple Aboodh transform, double Aboodh transform, Caputo fractional derivative, Summed transform method. 1. INTRODUCTION Many real world problems that arise in all the field of applied science are described by partial differential equation of integer and non-integer order. Many researchers have turned their attention to solve partial differential equation and to develop new methods for solving such equations, due to that many papers are published for developing methods for solving partial differential equation, integral equations, fractional differential equation and so on[1,2,3 ,5] "in press" [4]. One of the well-known methods for solving these equations is the integral transform methods, like Laplace transform method [6, 7], Summed transform method [8, 9], Natural transform [10], Ezaki transform method [11], and so on. Khalid Aboodh[12] in 2013 introduced a new integral transform called Aboodh transform, which is derived from the Fourier integral and similar to Laplace transform, and applied it to solve ordinary differential equation , after that he introduced the double Aboodh transform and used it to solve Integral differential equation and partial differential equation[13]. Aboodh transform method proved very affection methods to solve partial differential equation, and fractional differential equation. The objective of this article is to extend the Aboodh transform to the triple Aboodh transform, and discuss some theorems and properties about the triple Aboodh transform. To show the applicability and efficiency of this interesting transform we apply this transform to some test examples. First, we recall the definition of first Aboodh and double Aboodh transforms given by Khalid Aboodh [12] the definition of first Aboodh transform is given by:       0 1 ( ) , x>0, ptA f x K p f x e dt p      and the inverse Aboodh transform is defined by:   1 , 2 ( )K p dp       α i px α i f x y p e πi . The double Aboodh transform [13] is defined by         0 0 1 , , , , , , px qy x yA A f x y p q K p q f x y e dxdy pq           where  ,f x y is continuous function and , 0x y  . Moreover, the inverse of double Aboodh transform is given by:   ( , 1 1 , 2 2 )K p q dq dp                 β iα i px qy α i β i f x y p e q e πi πi This article has been organized as follows: In Section 2 we introduce the definition of triple Aboodh transform, and we present the triple Aboodh transform of some partial derivative of function of three variables, in section 3 we present existence and uniqueness of triple Aboodh transform. In section 4, we state the convolution theorem of the triple Aboodh transform and its proof. Some theorems and properties of the triple Aboodh transform method are given in section 5. In section 3, we give an analysis of the proposed method. In section 6, we demonstrate the applicability of the triple Aboodh transform by presenting three examples. Finally, the conclusion follows in section 7. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 42 2. DEfiNITION OF THE TRIPLE ABOODH TRANSFORM In this section, we introduce the definition of triple Aboodh transform and triple Aboodh transform of partial and fractional derivatives which are used further in this paper, moreover we apply triple Aboodh transform for some basic functions. Definition 2.1 let f be a continuous function of three variables, then the triple Aboodh transform of  , ,f x y t is defined by:          0 1 , , , , , ,x y t t         px qy st p q r A A A f x y t e f x y dxdydt pqr K (1) In addition, the inverse of triple Aboodh transform is given by:   1 1 1 ( , , 1 1 1 , , 2 2 2 )x y t K p q r dr dq dpt A A A                                β i γ iα i px qy rt α i β i γ i f x y p e q e r e πi πi πi (2) First of all, we find triple Aboodh transform for partial derivatives The triple Aboodh transform of nth derivative of a function of three variables is given by: 1 2 0 1 2 0 ( , , ) (0, , ) ( ( , , )) , ( , , ) ( ,0, ) ( ( , , )) , ( , , ) nn m n n m x y t x y t y tn m m nn m n n m x y t x y t x tn m m n n x y t n f x y t f y t A A A p A A A f x y t p A A x f x y t f x t A A A q A A A f x y t q A A y f x y t A A A r t                                                   1 2 0 ( , ,0) ( ( , , )) . n m n m x y t x y m m f x y A A A f x y t r A A              The triple Aboodh transform of mixed derivative of a function of three variables is given by: 3 ( , , , ) ( , , ) ( , ,0) ( ,0, ) 1 (0, , ) ( ,0,0) (0, ,0) (0,0, ) (0,0,0) x y t f x y z t pq pr A A A pqrK p q r K p q K p r x y t r q qr p q r K q r K p K q K r f p qr pr pq pqr                        3 2 2 2 2 ( , , ) ( , , , ) (0, , ) (0, , ) 1 (0, ,0) ( , ,0) 1 (0, ,0) x y t x y t y t y t x y y f x y t A A A rp A A A f x y z t rA A f y t t x r f y t p f y A A A A f x y A p x q pr x A f y r                           The triple Aboodh transform of the partial fractional Caputo derivatives of a function of three variables is given by: 1 2 0 1 2 0 ( , , ) (0, , ) ( ( , , )) , ( , , ) ( ,0, ) ( ( , , )) , ( , , ) j k k x y t x y t y t k k n k k x y t x y t x t k k x y t f x y t f y t A A A p A A A f x y t p A A x x f x y t f x t A A A q A A A f x y t q A A y y f x y t A A A t                                                             1 2 0 ( , ,0) ( ( , , )) . m k k x y t x y k k f x y r A A A f x y t r A A t               Where the Caputo fractional derivative [14] of function ( , , )f x y t defined by:  Aboodh transform of the Some Functions International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 43 a. If  , , 1f x y t  , for 0, 0, 0,x y t   then   2 2 2 1 , ,K p q r p q r  . b. If  , ,f x y t xyt , then   3 3 3 1 , , .K p q r p q r  c.         1 .ax by ctx y tA A A e p p a q q b r r c       d.   3 3 3 .8x y t A A A xyt p q r    e.       3 3 3 cos .x y t p q r pqr A A A x y t p p q q r r          f.      3 3 3 1 sin pq pr qr x y t p p q q r r           g.           2 2 2 2 2 2 1 .x y tA A A cos ax cos by cos ct p a q b r c    h.             2 2 2 2 2 2 sin sin sin .x y tA A A ax by ct ab p p a q q b c r r c    i.   2 2 2 ( 1) ( 1) ( 1) .m n vx y t m n v m n v A A A x y t p q r          3. EXISTENCE AND UNIQUENESS OF TRIPLE ABOODH TRANSFORM In this section we discuss the existence and uniqueness of the triple Aboodh transform, and we prove the uniqueness of the triple Aboodh transform. Let ( , , )f x y t be a continuous function on the interval  0, , which is of exponential order, that is for some , ,a b c  ( ) , , 0 ( , , ) sup ax by ct x y t f x y t e     . Under the above condition, the triple Aboodh transform exists for all , ,p a q b r c   . In the next theorem, the uniqueness of the triple Aboodh transform is proven. Theorem 3.1: let ( , , )h x y t and ( , , )l x y t be continuous functions defined for , , 0x y t  and having the Aboodh transform ( , , )H p q r and ( , , )L p q r respectively. If ( , , )H p q r = ( , , )L p q r , then ( , , )h x y t = ( , , )l x y t . Proof: If we assume , ,   to be sufficiently large, then since   1 1 1 , , 2 2 2 ( , , )K p q r dr dq dpt                            β i γ iα i px qy rt α i β i γ i f x y p e q e r e πi πi πi We deduce that International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 44         1 1 1 , , , , 2 2 2 1 1 1 , , 2 2 2 , , r dq dpt d dr dq t                                                      β i γ iα i px qy rt α i β i γ i β i γ iα i px qy rt α i β i γ i h x y p e q e r e H p q r πi πi πi p e q e r e L p q r πi πi πi = l x y and the theorem is established. Theorem 3.2: If  ( , , ) ( , , )x y tK p q r A A A f x y t , Then      , , , , , ,pa qbx y rc tA f x a y b t c H x a y bA A t c e K p q r            Where  , ,H x y t is the Heaviside unit step function defined by :   1 , , , , 0 , , x a y b t c H x a y b t c x a y b t c            Proof: By defnintion we have           0 , , , , 1 , , , , 1 , , px qy rt px qy rt x y t a b c A f x a y b t c H x a y b t c e f x a y b t c H x a y b t c dxdydt pqr e f x a y b t c dxdydt pqr A A                                 By letting 1 2 3 , , x a u y b u t c u           1 2 3 1 2 3 1 2 3 0 1 , , , , , , , pu qu rupa qb rc x y tA f x a y b t c H x a y b t c e e f u u u du du du pqr A A                  , , . pa qb rce K p q r   4. CONVOLUTION THEOREM In this section, we state and prove the convolution theorem of triple Aboodh transform. Theorem 4.1: If at the point ( , , )p q r the integral      11 0 1 , , , ,t        px qy rt p q r e g x y dxdydt p G qr is converge, and in addition if      22 0 1 , , , ,t        px qy rt p q r e g x y dxdydt p G qr is absolutely converge, then the following expression 1 2( , , ) ( , , ) ( , , )G p q r pqrG p q r G p q r is the Aboodh transform of the function 1 1 1 1 2 1 1 1 1 1 1 0 0 0 ( , , ) ( , , ) ( , , ) yt x g x y t g x x y y t t g x y t dx dy dt      and the integral International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 45       0 1 , , , ,t        px qy rt p q r e g x y dxdydt pqr G is converge at the point ( , , )p q r . Proof:         1 1 1 1 2 1 1 1 1 1 1 0 0 0 0 0 , , ( , , ) ( , , ) 1 , , 1 yt x x y t g x x y y t t g G p q r dxdyd x y t dx dy dt t pqr dxdydt pqr                           px qy rt px qy rt e g e By using Heaviside unit step function   2 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 ( , , ) ( , , ) ( , , )g x y t dx dy dt g x x y y t t H x x y y t t dxdydt pqr                       px qy rt e  1 1 1 12 1 1 1 1 1 1 0 1 2 ( , , ,) ( , ) , , ( , , ) px qy rt e G p qg x y t dx dy dt pqrG p q r G p q r r         5. Some properties of triple Aboodh transform In this section, we discuss and prove various properties of tripe Aboodh transform.  The triple Aboodh transform is a linear operator, that is       ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) x y t x y t x y t A A A af bg x y t p q r aA A A f x y t p q r bA A A g x y t p q r      Proof:           0 0 0 1 1 1 . ( , , ) ( , , ) ( , , ) ( , , ) ). ( , , x y t pqr a b pqr A A A af bg x y t p q r af bg x y t dxdydt f x y t dxdydt g x y t dxd q d r y p t                         px qy rt px qy rt px qy rt e e e    . ( , , ) ( , , ) . ( , , ) ( , , )x y t x y ta A A A f x y t p q r b A A A g x y t p q r   Changing of scale property: If  ( , , ) ( , , )x y tA A A f x y t K p q r , then   1 ( , , ) , ,x y t p q r A A A f ax by ct K abc a b c        Proof:     ( 0 ) 0 0 0 0 ( , , ) ( , , ) 1 1 ( , , ) 1 1 , , 1 1 , , x y t p a x qy rt px A A A f ax by ct f ax by ct dxdydt e e f ax by ct dydt dx p qr q r e K x dx p bc b pq c p q r K abc a b c r                                           px qy rt e Note that the first and double Aboodh transforms satisfy the changing of scale property. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 46  Shifting property: If  ( , , ) ( , , )x y tA A A f x y t K p q r , then ( , , ) ( , , ) ax by ct x y tA A A e f x y t K p a q b r c           Proof:   ( ) ( ) ( ) 0 0 0 ( ) 0 0 ( , , ) ( , , ) 1 1 ( , , ) 1 ( , , ) ( , 1 , ) ax by ct ax by ct x y t p a x q b y r c t p a x A A A e f x y t e f x y t dxdydt e e f x y t dydt dx p qr e K x q b r c dx p K p a q b r c pqr                                                 px qy rt e Note that first and double Aboodh transforms satisfy the shifting property.  Multiplying by n m vx y t   ( 1) ( , , ) ( , , ) n m v n m v n m v x y t n m v A A A x y t f x y t pqrK p q r pqr p q r            Proof:   0 0 0 0 1 ( , , ) ( , , ) 1 1 ( , , ) n m v n m v x y t n px qy rt m v dA A A x y t f x y t x y t f x y t xdydt pqr x e e y t f x y t dt dx q y p d r                          px qy rt e The expression in the bracket satisfies the property of the double Aboodh transform, that is   ( 1) ( , , ) ( , , ) m v m v m v y t m v A A y t f x y t qrK x q r qr q r        , thus,     0 1 ( 1) ( , , ) ( 1) ( , , ) , m v m v n px m v n m v n m v n m v x e qrK x q r dx p qr q r pqrK p q r pqr p q r                and this complete the proof.  If ( , , ) ( ) ( ) ( )f x y t g x h y z t , then        ( , , ) ( ) ( ) ( )x y t x y tA A A f x y t A g x A h y A z t . Proof: International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 47             0 0 0 0 0 1 ( , , ) ( , , ) 1 ( ) ( ) ( ) 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) x y t px qy rt x y t A A A f x y t f x y t xdydt pqr g x h y z t xdydt pqr e g x dx e h y dy e z t dt p q r A g x d d A h y A z t                                              px qy rt px qy rt e e Theorem 5.1: An exponentially of order continuous function ( , , )f x y t on  0, can be recovered from only ( , , )K p q r as: 11 1 , , ( 1) ( , , ) lim ( , , ) ! ! ! mn vn m v n m v n m vn m v n m v n m v f x y t pqrK n m v x y t x y tp q r                                . Proof: The poof is similar to proof given by Abdon Atangana see [15]. To check the efficiency of the previous theorem, let us consider the following example: Let ( , , ) ax by ctf x y t e   for which the triple Aboodh transform can be found as: 1 ( , , ) ( ) ( ) ( 1) K p q r p p a q q b r r     , by applying the high-order mixed derivative to the expression, we get the following:   1 1 1 ! ! !( 1) ( , , ) ( ) ( ) ( ) n m v n m v n m v n m v n m v pqrK p q r p q r p a q b r c                 , by using theorem( 5.1), we get: 1 11 1 1 1 , , 1 1 1 , , ( , , ) lim lim 1 1 1 , m mn v n v n m v n m v n m v n m v n m v f x y t a b c x y t x y t ax by ct n m v                                                                          by applying the logarithm and L'Hopital's rule we get: ln ( , , ) , ( , , ) .ax by ct f x y t ax by ct f x y t e         6. APPLICATION In this section, we construct some different examples to illustrate the applicability and efficiency of the triple Aboodh transform. Example 1: Consider the following fractional partial differential equation: 2 2 ( , , ) ( , , ) , 0 1, (3) t f x y t D f x y t x        , with the following initial and boundary values: (0, , ) 0, (0, , ) sin( ) ( ) ( , ,0) sin( )sin( ) xf y t f y t y E t f x y x y      Solution: Applying the triple Aboodh transform to equation (3) and for the initial and boundary conditions we get: International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 48         2 2( , , ) ( , ,0) ( , , ) 1 (0, , ) (0, , ) , x y t x y x y t y t y t r A A A f x y t r A A f x y p A A A f x y t f y t A A f y t A A p x            Where,       2 1 1 ( , ,0) sin sin ( 1) ( 1) (0, , ) (0) 0 (0, , ) 1 , ( 1)1 x y x y y t y t y t A A f x y A A x y p p q q A A f y t A A f y t r A A x q qr                 thus,         2 2 2 2 2 22 1 1 ( , , ) ( 1) 1 1 1 1 ( , , ) , 1 ( 1)1 x y t x y t r r p A A A f x y t pq q p r r A A A f x y t r q qp p                      by applying the inverse of triple Aboodh transform, we get:   2 1 1 1 22 1 1 ( , , ) sin( )sin( ) ( ) 1 ( 1)1 x y t r f x y t A A A x y E t r q qp p                    . Example 2: Consider the following nonhomogeneous third-order Mboctara partial differential equation: 3 2( , , ) ( , , ) 3 , (4)x y t f x y t f x y t e x y t         subject to the following initial and boundary conditions: 2 2(0, , ) , ( ,0, ) , ( , ,0)y t x t x yf y t e f x t e f x y e        . Solution: Applying the triple Aboodh transform to equation (4), we get: 1 ( 1) ( , , ) ( , , ) , (5) ( 1) ( 2) ( 1) pqr K p q r U p q r p p q q r r       Where, ( , , ) ( , ,0) ( ,0, ) (0, , ) 1 ( ,0,0) (0, ,0) (0,0, ) (0,0,0) 2 , ( 1) ( 2) ( 1) pq pr qr U p q r K p q K p r K q r r q p p q r K p K q K r f qr pr pq pqr pqr p p q q r r             By substituting the value of ( , , )U p q r in equation (5), we get: 1 ( , , ) ( 1) ( 2) ( 1) K p q r p p q q r r     , by applying the triple inverse Aboodh transform we get: 2( , , ) .x y tf x y t e    Example 3: Consider the following diffusion equations: 2 2 2 2 ( , , ) ( , , ) ( , , ) 5 0, (6) f x y t f x y t f x y t tx y         Subject to the following initial and boundary conditions: International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 49 2 1 2 2 1 2 (0, , ) , (1, , ) ( ,0, ) , ( ,0.5, ) ( , ,0) , y t y t x y x x y f y t e f y t e f x t e f x t e f x y e            Solution: By applying the triple Aboodh transform to equation(6), we get:  2 2 ( , , ) ( , , ) (7)p q r K p q r U p q r   Where       2 2 1 (0, , ) ( , , ) (0, , ) 1 ( ,0, ) 1 ( ,0, ) ( , ,0) ( 1) ( 2) ( 1) y t y t x t y t x y f y t U p q r A A f y t A A p x f x t A A f x t A A A A f x y q y r p q r p p q q r r                       By substituting the value of ( , , )U p q r , equation (7) becomes: 1 ( , , ) (8) ( 1) ( 2) ( 1) K p q r p p q q r r     Applying the triple inverse Aboodh transform on equation(8), we get : 7. CONCLUSION In this present work, triple Aboodh transform and its inverse are defined in order to solve partial differential equations and fractional differential equations, furthermore, we present several properties and theorems of triple Aboodh transform. To see the efficiency of triple Aboodh transform, we apply this transform on three different examples; the results show that the triple Aboodh transform method is an appropriate method for solving partial differential equations of both integer and fractional order. REFERENCES [1] A.K. Hassan Sedeeg, M.M. Abdelrahim Mehgoub, Aboodh Transform of Homotopy Perturbation Method for Solving System of Nonlinear Partial Differential Equations, Math. Theo. and Mod., 6(8), 108-113, 2016. Maxwell, J. C. (1892). A Treatise on Electricity and Magnetism, 3rd ed., vol. 2. Oxford: Clarendon, pp.68-73. [2] S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992. [3] Ogunfiditimi F.O. (2015). Numerical solution of delay differential equations using Adomian decomposition method. The International Journal of Engineering and Science, 4(5): 18 – 23 [4] S. Alfaqeih, T.OZIS Solution of Black-Scholes Fractional Partial Diff erential Equation with two Assets by Aboodh Decomposition, Method Progr. Fract. Differ. Appl.1, No. 1, 1-12 (2019) [5] KeskinY.Oturan G., Reduced diff erential transform method for generalized Kdvequations,Mathematical and Computational applications, 15 (3), 382-393 ,2010. [6] V. G. Gupta, P. Kumar, Approximate solutions of fractional linear and nonlinear differential equations using Laplace homotopy analysis method, Int. J. Nonlinear Sci.,19(2) (2015), 113-12 [7] Kili'cman A. A., Eltayeb H. and Atan K.A.M., A note on the comparison between Laplace and Sumudu transforms, Bull Iranian Math Soc, 37 (2011) 131-141 [8] V. G. Gupta, P. Kumar,Approximate solutions of fractional biological population model by homotopy analysis Sumudu transform method, Int. J. Sci. Res.,5(5) (2016), 908-917. [9] Belgacem F.B.M. and Karaballi A.A., Sumudu transform fundamental properties investigations and applications, Journal of applied mathematics and stochastic analysis, (2006). 2( , , ) .x y tf x y t e   International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2000-000X Vol. 3 Issue 3, March – 2019, Pages: 41-50 www.ijeais.org 50 [10] Khan Z.H., and Khan W.A., Natural transform-properties and applications, NUST Journal of Engineering Sciences, 1(2008) 127-133Young, M. (1989). The Technical Writer's Handbook. Mill Valley, CA: University Science. [11] Elzaki T.M., The new integral transform "Elzaki transform", Global Journal of Pure and Applied Mathematics, 7(1) (2011) 57-64. [12] K.S. Aboodh, The new integral transform "Aboodh transform" Global Journal of Pure and Applied Mathematics, 9(1) (2013) 35-43. [13] K.S. Aboodh , R.A. Farah , I.A. Almardy and F.A. ALmostafa Solution of Partial Integro-Differential Equations by using Aboodh and Double Aboodh Transform Methods, Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 8 (2017), pp. 4347-4360. [14] A.A. Kilbas, H.H. Srivastava, J.J.Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam. 2006. [15] AbdonAtangana A Note on the Triple Laplace Transform and Its Applications to Some Kind of Third-Order Differential Equation Abstract and Applied Analysis Volume 2013,Article ID769102. Authors Suliman Alfaqeih, obtained his Bachelor's degree in Mathematics from Al-Quds University in 2009 and Masters in Applied mathematics in 2011 from the same University, from Palestine. In 2011 he worked as a lecturer in mathematics in al-Quds open university in Palestine After that he moved to Saudi Arabia and worked as a lecturer in mathematics in king Saud university from 2012 to 2016. In 2016, he got the Turkish scholarship and moved to turkey, presently he is a PhD student at Ege University. Current research interests include numerical methods of fractional differential equations, integral transform method of partial and fractional differential equations. Turgut Ozis    obtained his B.S. degree in Mathematics from Ege University, Izmir, Turkey, in 1973. He received his M.S. and Ph.D. degrees from Brunel University, London, U.K. on free and moving boundary problems in 1981. In late 1982, he moved to what was then Inonu University, Malatya, Turkey, and worked as the Head of the Mathematics department until 1991. After that he was persuaded to accept an Associate Professorship position in the Science Department of Ege University. In 1992, he was appointed as a Professor of Applied Mathematics at Ege University, since he had been the Head of the Applied Mathematics Group. His research interests include numerical methods for partial differential equations (both linear and nonlinear), free boundary problems, reaction diffusion problems, numerical heat transfer, two-point boundary value problems for ordinary differential equations, and exact approximate methods for nonlinear systems. Prof. Ozis  is a member of the editorial boards of three international journal..