Council for Innovative Research

Transcription

1 ISSN: A solution of Fractional Laplace's equation y Moifie separation of variales ABSTRACT Amir Pishkoo,, Maslina Darus, Fatemeh Tamizi 3 Physics an Accelerators Research School (NSTRI) P.O. Box , Tehran, Iran apishkoo@gmail.com (corresponing author) School of Mathematical Sciences, Faculty of Science an Technology Universiti Keangsaan Malaysia, Bangi, Selangor Darul.Ehsan, Malaysia maslina@ukm.my 3 Islamic Aza University Zahean Branch, Zahean, Iran fatemehtm@gmail.com This paper applies the Moifie separation of variales metho (MSV) suggeste y Pishkoo an Darus towars otaining a solution for fractional Laplace's equation. The close form expression for potential function is formulate in terms of Meijer's G-functions (MGFs). Moreover, the relationship etween fractional imension an parameters of Meijer s G- function for spherical Laplacian in R, for raial functions is erive. Keywors: Fractional ifferential equation; Laplace's equation; Meijer's G-function; Moifie separation of variales. Council for Innovative Research Journal: Journal of Avances in Physics Peer Review Research Pulishing System Vol 4, No. eitor@cirworl.com memer.cirworl.com 397 P a g e M a r c h 0 5, 0 4

2 INTRODUCTION ISSN: It is a known fact that shapes an forms occur in nature which cannot e escrie y Eucliean geometry. Examples of these inclue, ranching of trees, the roughness of ocean floor, the geometry of clous, the ust in planetary isks etc. All of these are instances where high spatial complexity occurs. Such structures are not easily escriale ecause we cannot assign a characteristic length to them []. Manelrot [, ] first introuce the term, "Fractal" to ifferentiate pure geometries from other types which o not fit into a simple classification. Fractal is use to escrie the egree of irregularity or fragmentation of a sample or structure which is ientical at all scales. In the fractal meia, some of the regions or omains are not fille with meium particles an these unfille omains are calle porous omains. Fractal moels of meia are ecoming increasingly popular ecause just a small numer of parameters are neee to efine a meium of great complexity an rich structure. In orer to use fractional erivatives an fractional integrals for meia on fractal, we must use some continuous moel [, 3-4]. Most times, the real fractal structure can e isregare an the meium on fractal can e escrie y some fractional continuous mathematical moel. The orer of the fractional integral is equal to the fractal imension. Fractional integrals can e consiere as approximations of integrals on fractals [5, 6]. Fractals are characterize y a fractional imension,. Hence complex structures can e moele at microscopic an macroscopic levels [7, 8]. The concept of fractional imensional space has wiesprea applications in several areas of physics, essentially to escrie the effective parameters of physical systems [3, 9]. It is important to apply a generalization of electromagnetic theory in fractional space in orer to extract full enefits of fractal moels, which are ecoming popular ue to small numer of parameters that efine a meium of greater complexity an a rich structure. Fractional calculus is use y ifferent authors to escrie fractional solutions to many electromagnetic prolems as well as fractional imensional space [0-]. It was propose that smoothing of microscopic characteristics over the physically infinitesimal volume transforms the initial fractal istriution into fractional continuous moel that uses fractional integrals [3]. Solutions to Poisson's an Laplace equation for scalar potential in fractional space have een iscusse in [4, 5]. Stillinger [6] has evelope the formalism for constructing a generalization of an integer imensional Laplacian operator into a non-integer imensional space. Laplace's equation in fractional space is erive y using moifie vector ifferential operators: Plamer an Stavrinou [9] generalize the Stillinger's results to n orthogonal coorinates an Laplacian operator in imensional fractional space in three-spatial coorinates is given as: where, parameters are use to escrie the measure istriution of space where each one is acting inepenently on a single coorinate an the total imension of the system is On the other han, forms of Laplacian in R, R 3, an R n (n is integer) imensional space in three coorinates systems are as follows [8]: Polar Laplacian in R for Raial functions Spherical Laplacian in R3 an Rn for Raial Functions (.) (.) (.3) (.4) Cylinrical Laplacian in R3 for Raial functions Three analytical solution methos are use to solve ifferential equations namely; Green's function metho, Variational metho, an Separation of variales. Physically, the Green's function represents the fiel prouce y a point source. By using the Green's function, the solution of Maxwell equations can e represente y an integral efine over the source region or on a close surface enclosing the source. Mathematically, the solution of a partial ifferential equation, where stans for the inverse of can e expresse as an is often represente y an integral operator whose kernel is the Green's function. The principle of least action leas to the evelopment of the Lagrangian an Hamiltonian formulations of classical mechanics. Base on the principle of least action, the ifferential equations of a (.5) (.6) 398 P a g e M a r c h 0 5, 0 4

3 ISSN: given physical system are erivale y minimizing the action of the system. The original prolem, governe y the ifferential equations, is thus replace y an equivalent variational prolem [8]. The metho of separation of variales (SV) is usually use to seek a solution of the form of the prouct of functions, each of which epens on one variale only, so that the solution of original partial ifferential equations may reuce to the solution of orinary ifferential equations [9]. In this paper the newly metho calle Moifie separation of variales (MSV) is use to solve fractional Laplace's equation. Our previous work ha focuse on introuction of the Moifie separation of variales metho (MSV), an applying it to solve partial ifferential equation relate to the Reaction-Diffusion process, Laplace s equation, an Schroinger equation which leas to represent its solution in terms of Meijer's G-functions [7-0]. In [] we iscuss the fractional transformations of Meijer's G-functions y illustrating how G-functions are more general in form than hypergeometric functions. This paper uses the MSV metho to solve fractional Laplace's equation an finally it euces the solution in terms of Meijer's G-functions. The paper has een organize as follows. Section gives notation an asic efinition of Meijer's G-function. Section 3 introuces the Moifie separation of variales (MSV) suggeste y Pishkoo an Darus in [7]. Section 4 consists of main results of the paper, in which the Moifie separation of variales metho has een evelope for fractional Laplace's equation. MEIJER G-FUNCTIONS In mathematics, the G-function was introuce y Cornelis Simon Meijer (936) as a very general function intene to inclue all elementary functions an most of the known special functions as particular cases, for instance: A efinition of the Meijer's G-function is given y the path integral in the complex plane, calle Mellin- Barnes type integral see []-[6]. For the G-function The integers m; n; p; q are calle orers of the G-function, ap an q are calle "parameters" an in general, they are complex numers. The efinition hols uner the following assumptions: (.7) for functions, where are integer numers. For example the orers,0,0, are the same ; ut their parameters are ifferent. The Meijer's G-function satisfies the linear orinary ifferential equation of the generalize hypergeometric type whose orer is equal to max (p, q). (See [], [7-9]) p q p m n [( ) z ( z a ) ( z k)] y ( z ) = 0. z (.8) j j= z k = The funamental question is that why o we suggest working with G-functions espite their complex forms? The first reason is that each of G-functions immeiately has ifferential equation (8) on their sie. For instance when we iscuss aout function there is no image of its ifferential equation, 399 P a g e M a r c h 0 5, 0 4

4 ISSN: while for we can easily otain its ifferential equation. The secon reason will e shown when we use the following Moifie Separation of Variales metho to solve partial ifferential equations or linear orinary ifferential equations. MODIFIED SEPARATION OF VARIABLES (MSV) METHOD [7] In solving Partial Differential Equations (PDE) initial an ounary conitions are necessary to otain specific solutions, an epening on the geometry of the prolem, these ifferential equations are separate into Orinary Differential Equation (ODEs) in which each involves a single coorinate of a suitale coorinate system. With Cartesian, cylinrical an spherical coorinates, the ounary conitions are important in etermining the nature of the ODE solutions otaine from the PDE. The Separation of Variales is employe to otain solutions to the many partial ifferential equations. The metho consists of assuming the solution as a prouct of functions, each epening on one coorinate variale only. The metho propose in [7], i.e. the Moifie separation of variales, leas to otain the solution as a prouct of the Meijer's G- functions, with each one epening on one variale y equating orinary linear ifferential equation of the Meijer's G- functions an each one of the ODEs. Four steps are followe in this metho, namely: ) Choosing a convenient coorinates system y consiering the ounary conitions an ounary surfaces. ) Otaining ODEs from PDE y writing the solution as a prouct form of ifferent variales an putting it into PDE. 3) Starting with (8), the orers m, n, p, an q, are chosen an the variale z is change to such that Eq.(8) converts into each of ODEs, an this is three times for time inepenent prolem (e.g., x, y, z coorinates) an four times for time epenent prolem (e.g., x, y, z, t). If this particular step is one, solving ODEs is not require ecause Equation (8) is an equation that oes not nee to e solve. It is solve, an the solutions are MGFs. 4) Using ounary conitions an otaining the exact solution. MAIN RESULTS- Form of Polar Laplacian in For each >0 r u = u( x ( r, ), y ( r, )) x ( r, ) = r cos, y ( r, ) = r sin u = u cos u sin, u = u r sin u r cos. r x y x y u tan = u x = ( )[ u r u ]. x x x r r (.9) u cot = u y = ( )[ u r u ]. y y y r r (.0) From an u = u u cos sin u rr xx cos xy yy sin u r u u u r u u = ( xx sin xy cos sin yy cos ) ( x cos y sin ). We have u u u u u =. x y r r r r (.) Comining (9), (0), an (), we otain: 400 P a g e M a r c h 0 5, 0 4

5 ISSN: u u u u u ( ) u tan u = ( )( ) x x x y y y r r r r (.) cot u ( )( u ). r r Finally Polar Laplacian in R u for Raial function ( u = u ( r), = 0), where, is as follows: u u u u u ( ) u =. x x x y y y r r r (.3) MGF s solution for fractional Laplace's equation in We here consier two cases when (8) is reuce to first an secon orer orinary ifferential equation, respectively: Simplifying gives CASE. Setting m =, n = 0, p =, q = in (8) yiels,0 a [ z ( z a )] ( z )] G, ( z ) = 0. z z,0 a, ( ) ( ) G z = z a z, z z ( z ) Now putting a = = c leas to the solution z c CASE. Setting m =, n =, p =, q = in (8) yiels By changing z to z-, we have,0 ( c ) = c G., z c z, a, a [ z ( z a )( z a ) ( z )( z )] G, ( z, ) = 0. z z z z, a, a [ ( z )[( z ) a ][( z ) a ] [( z ) ][( z ) ] G, ( z, ) = 0. z z z z (.4) Comparing aove equation with the form of Polar Laplacian in R or Cylinrical Laplacian in R 3 for Raial functions, namely [ ] ( ) = 0. uz z z z Equality conition for these two ifferential equations leas to a =a = =, =0, an the solution in terms of Meijer's G- function is u( z ) = G ( z ) = ln z. (.5),,,,0 Moreover, comparing again ut this time with the form of Spherical Laplacian in R 3 for Raial Functions namely (4), we otain parameters of G-function as follows: An the solution is a a =, =, = P a g e M a r c h 0 5, 0 4

6 ISSN: , a, a,,0 u( z ) = G ( ) (.6) Thus general solution gives us, a, a,0 c,0 c,,0, c, c u( z ) = A BG ( ) CG ( z ) DG ( z ), MSVmetho, (.7) Where A, B, C, D are constants, an they can e etermine y ounary conitions. Finally an most significantly we otain the relation etween fractional imension in general an parameters of G- functions for Spherical Laplacian in R D for Raial Functions, namely (5) ACKNOWLEDGMENTS =α +α =+a -a, =, =0. (.8) This work was supporte y MOHE with the grant numer: ERGS//03/STG06/UKM/0/. REFERENCES [] Asa, H., Zuair, M. an Mughal, M.-J. Reflection an Transmission at Dielectric-Fractal interface, Progress in Electromagnetics Research 5 (0) pp [] Manelrot, B. The Fractal Geometry of Nature, W. H. Freeman, New York, 983. [3] Tarasov, V. E. Electromagnetic fiels on fractals, Moern Physics Letters A (006) pp [4] Tarasov, V. E. Continuous meium moel for fractal meia, Phys. Lett. A 336 (005) pp [5] Svozil, K. Quantum fiel theory on fractal spacetime: a new regularization metho, J. Phys. A 0 (987) pp [6] Ren, F. Y., Liang, J. R., Wang, X. T. an Qiu, W. Y. Integrals an erivatives on net fractals, Chaos, Soliton an Fractals 6 (003) pp [7] Vicsek, T. Fractal moels for iffusion controlle aggregation, J. Phys. A: Math. Gen. 6 (983) pp [8] Wagner, G. C., Colvin, J. T., Allen, J. P. an Stapleton, H. J. Fractal moels of protein structure, ynamics an magnetic relaxation, J. Am. Chem. Soc. 07 (985) pp [9] Palmer, C. an Stavrinou, P. N. Equations of motion in a non-integer-imension space, Journal of Physics A 37 (004) pp [0] Hussain, A., S. Ishfaq, Q. A. Naqvi, Fractional curl operator an fractional waveguies, Progress in Electromagnetics Research Letters, PIER 63 (006) pp [] Zuair, M., Mughal, M. J. an Naqvi, G. A. Rizvi, Differential electromagnetic equations in fractional space, Progress in Electromagnetics Research Letters, PIER 4 (0) pp [] Zuair, M., Mughal, M. J. an Naqvi, G. A. An exact solution of the cylinrical wave equation for electromagnetic fiel in fractional imensional space, Progress in Electromagnetics Research Letters, PIER 4 (0) pp [3] He, J. H. Approximate analytical solution for seepage flow with fractional erivatives in porous meia, Comput. Methos Appl. Mech. Eng. 67 (998) pp [4] Muslih, S. I. an Baleanu, D. Fractional multipoles in fractional space, Nonlinear Analysis: Real Worl Applications 8 (007) pp [5] Baleanu, D., Golmankhaneh, A. K. On electromagnetic fiel in fractional space, Nonlinear Analysis: Real Worl Applications (00) pp [6] Stillinger, F. H. Axiomatic asis for spaces with non-integer imension, Journal of Mathematical Physics 8 (977) pp [7] Pishkoo, A. an Darus, M., Meijers G-functions (MGFs) in Micro- an Nano-structures, Journal of Computational an Theoretical Nanoscience 0 (03) pp [8] Pishkoo, A. an Darus, M., Some Applications of Meijer G-functions as Solutions of Differential Equations in Physical Moels, Journal of Mathematical Physcis, Analysis, Geometry, 3 (03) pp [9] Pishkoo, A. an Darus, M., G function solutions for Diffusion an Laplace's equations, Journal of Avances in Mathematics, 4 (03) pp P a g e M a r c h 0 5, 0 4

7 ISSN: [0] Pishkoo, A. an Darus, M., G-function solutions for Schroinger equation in Cylinrical coorinates system, Applie Mathematics, 5 (04) pp [] Pishkoo, A. an Darus, M. Fractional ifferintegral transformation of univalent Meijers G-functions. Journal of Inequalities an Applications 0 (0) article 36. [] Kiryakova, V. Generalize Fractional Calculus an Applications. Longman, Harlow, UK (994). [3] Anrews, L. C. Special Functions for Engineers an Applie Mathematicians, MacMillan, New York (985). [4] Luke, Y. L. The Special Functions an Their Approximations, Complex Variales, vol. I. Acaemic Press, New York (969). [5] Klimyik, A. U. Meijer G-function. In Michiel, H (e.), Encyclopaeia of Mathematics, Springer, Berlin (00). [6] Askey, R. A. Meijer G-function. In Ari, D, Ole, B (es.) NIST Hanook of Mathematical Functions, Camrige University Press, Camrige (00). [7] Kilas, A. A., Srivastava, H. M. an Trujillo, J. J. Theory an Application of Fractional Differential Equations, Elsevier, Amsteram (006). [8] Prunikov, A. P., Brychkov, I. U. A. an Marichev, O. I. Integrals an Series: Some More Special Functions. Goron & Breach, New York (99). [9] A. Erelyi, A., Magnus, W., Oerhettinger an Tricomi, F. Higher Transcenental Functions, Vols. -3. McGraw- Hill, New York (953). [30] Geyi, W. Founation of Applie Electroynamics. Wiley (00). [3] Farlow, S. J. Partial Differential Equations for Scientists an engineers, Wiley (993). 403 P a g e M a r c h 0 5, 0 4