[Aggarwal, 5(8): Augus 2018] ISSN 2348 8034 DOI- 10.5281/zenodo.1339350 Impac Facor- 5.070 GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES ELZAKI TRANSFORM OF BESSEL S FUNCTIONS Sudhanshu Aggarwal Assisan Professor, Deparmen of Mahemaics, Naional P.G. College Barhalganj, Gorakhpur-273402, U.P., India ABSTRACT In he modern ime, Bessel s funcions appear in solving many problems of sciences and engineering ogeher wih many equaions such as hea equaion, wave equaion, Laplace equaion, Schrodinger equaion, Helmholz equaion in cylindrical or spherical coordinaes. In his paper, we deermine Elzaki ransform of Bessel s funcions. Some applicaions of Elzaki ransform of Bessel s funcions for evaluaing he inegral, which conain Bessel s funcions, are given. Keywords: Elzaki ransform, Convoluion heorem, Inverse Elzaki ransform, Bessel funcion. I. INTRODUCTION Bessel s funcions have many applicaions [3] o solve he problems of mahemaical physics, acousics, radio physics, aomic physics, nuclear physics, engineering and sciences such as flux disribuion in a nuclear reacor, hea ransfer, fluid mechanics, vibraions, hydrodynamics, sress analysis ec. Bessel s funcion of order, where is given by [1-5,10] In paricular, when Bessel s funcion of zero order and i is denoed by and i is given by he infinie power series Equaion h For Bessel s funcion of order one and i is denoed by and i is given by can be wrien as For Bessel s funcion of order wo and i is denoed by and i is given by The Elzaki ransform of he funcion is defined as [6]: where is Elzaki ransform operaor. The Elzaki ransform of he funcion for exis if is piecewise coninuous and of exponenial order. These condiions are only sufficien condiions for he exisence of Elzaki ransform of he funcion. Elzaki e al. [7] defined fundamenal properies of Elzaki ransform ogeher wih applicaions. HwaJoon Kim [8] gave he ime shifing heorem and convoluion for Elzaki ransform. Elzaki and Ezaki [9] discussed he connecions beween Laplace & Elzaki ransforms. Elzaki and Ezaki [12] used Elzaki ransform for solving ordinary differenial 45
[Aggarwal, 5(8): Augus 2018] ISSN 2348 8034 DOI- 10.5281/zenodo.1339350 Impac Facor- 5.070 equaion wih variable coefficiens. The soluion of parial differenial equaions using Elzaki ransform was given by Elzaki and Ezaki [13]. Shendkar and Jadhav [14] used Elzaki ransform for he soluion of differenial equaions. The objec of he presen sudy is o deermine Elzaki ransform of Bessel s funcions and explain he advanage of Elzaki ransform of Bessel s funcions for evaluaing he inegral which conain Bessel s funcions. II. LINEARITY PROPERTY OF ELZAKI TRANSFORM: If and hen where are arbirary consans. III. ELZAKI TRANSFORM OF SOME ELEMENTARY FUNCTIONS [6, 7] S.N. 1. 2. 3. 4. 5. Γ 6. 7. 8. 9. 10. IV. If CHANGE OF SCALE PROPERTY OF ELZAKI TRANSFORM: hen Pu in equaion Thus, if hen V. ELZAKI TRANSFORM OF THE DERIVATIVES OF THE FUNCTION [8, 14]: If hen 46
[Aggarwal, 5(8): Augus 2018] ISSN 2348 8034 DOI- 10.5281/zenodo.1339350 Impac Facor- 5.070 a) b) VI. CONVOLUTION OF TWO FUNCTIONS [11] Convoluion of wo funcions and is denoed by and i is defined by VII. CONVOLUTION THEOREM FOR ELZAKI TRANSFORM [7, 8] If and hen VIII. INVERSE ELZAKI TRANSFORM If hen is called he inverse Elzaki ransform of and mahemaically i is defined as where IX. is he inverse Elzaki ransform operaor. INVERSE ELZAKI TRANSFORM OF SOME ELEMENTARY FUNCTIONS S.N. 1. 2. 3. 4. 5. 6. Γ 7. 8. 9. 10. 47
[Aggarwal, 5(8): Augus 2018] ISSN 2348 8034 DOI- 10.5281/zenodo.1339350 Impac Facor- 5.070 X. RELATION BETWEEN AND [4, 10] h XI. RELATION BETWEEN AND [10] XII. ELZAKI TRANSFORM OF BESSEL S FUNCTIONS a) ELZAKI TRANSFORM OF : Taking Elzaki ransform of equaion, boh sides b) ELZAKI TRANSFORM OF : Taking Elzaki ransform of equaion h, boh sides Now applying he propery, Elzaki ransform of derivaive of he funcion on equaion Using equaion and equaion in equaion we have c) ELZAKI TRANSFORM OF Taking Elzaki ransform of equaion, boh sides Now applying he propery, Elzaki ransform of derivaive of he funcion and using equaion in equaion, we have Using equaion equaion h and equaion in equaion we have Using equaion (3) in equaion (17) 48
[Aggarwal, 5(8): Augus 2018] ISSN 2348 8034 DOI- 10.5281/zenodo.1339350 Impac Facor- 5.070 d) ELZAKI TRANSFORM OF : From equaion, Elzaki ransform of is given by Now applying change of scale propery of Elzaki ransform h e) ELZAKI TRANSFORM OF From equaion, Elzaki ransform of is given by Now applying change of scale propery of Elzaki ransform f) ELZAKI TRANSFORM OF From equaion, Elzaki ransform of is given by Now applying change of scale propery of Elzaki ransform XIII. APPLICATIONS In his secion, some applicaions are given in order o demonsrae he effeciveness of Elzaki ransform of Bessel s funcions for evaluaing he inegral which conain Bessel s funcions. APPLICATION:1 Evaluae he inegral Applying he Elzaki ransform o boh sides of Using convoluion heorem of Elzaki ransform on Operaing inverse Elzaki ransform on boh sides of 49
[Aggarwal, 5(8): Augus 2018] ISSN 2348 8034 DOI- 10.5281/zenodo.1339350 Impac Facor- 5.070 which is he required exac soluion of. APPLICATION:2 Evaluae he inegral Applying he Elzaki ransform o boh sides of Using convoluion heorem of Elzaki ransform on Operaing inverse Elzaki ransform on boh sides of h which is he required exac soluion of. APPLICATION:3 Evaluae he inegral Applying he Elzaki ransform o boh sides of Using convoluion heorem of Elzaki ransform on Operaing inverse Elzaki ransform on boh sides of which is he required exac soluion of. XIV. CONCLUSION In his paper successfully discussed he Elzaki ransform of Bessel s funcions. The given applicaions show ha he advanage of Elzaki ransform of Bessel s funcions o evaluae he inegral which conain Bessel s funcions. 50
[Aggarwal, 5(8): Augus 2018] ISSN 2348 8034 DOI- 10.5281/zenodo.1339350 Impac Facor- 5.070 REFERENCES 1. Mclachlan, N.W. Bessel funcions for engineers, Longman, Oxford (1955). 2. Bell, W.W. Special funcions for scieniss and engineers, D. Van Nosrand Company LTD London. 3. Korenev, B.G. Bessel funcions and heir applicaions, Chapman & Hall/CRC. 4. Farrell, O.J. and Ross, B.Solved problems in analysis: As applied o Gamma, Bea, Legendre and Bessel funcion, Dover Publicaions Inc. Mineola, New York. 5. Wason, G.N. A reaise on he heory of Bessel funcions, Cambridge Universiy Press, Cambridge (1944). 6. Elzaki, T.M.The new inegral ransform Elzaki Transform, Global Journal of Pure and Applied Mahemaics, 1, pp. 57-64, (2011). 7. Elzaki, T.M., Ezaki, S.M. and Elnour,E.A. On he new inegral ransform Elzaki Transform fundamenal properies invesigaions and applicaions, Global Journal of Mahemaical Sciences: Theory and Pracical, 4(1), pp. 1-13, (2012). 8. HwaJoon Kim The ime shifing heorem and he convoluion for Elzaki ransform, Inernaional Journal of Pure and Applied Mahemaics, 87(2), pp. 261-271, (2013). 9. Elzaki, T.M. and Ezaki, S.M.On he connecions beween Laplace and Elzaki ransforms, Advances in Theoreical and Applied Mahemaics, 6(1), pp. 1-11, (2011). 10. Raisinghania, M.D.Advanced differenial equaions, S.Chand & Company PVT LTD Ramnagar, New-Delhi. 11. Lokenah Debnah and Bhaa, D.Inegral ransforms and heir applicaions, Second ediion, Chapman & Hall/CRC (2006). 12. Elzaki, T.M. and Ezaki, S.M.On he Elzaki ransform and ordinary differenial equaion wih variable coefficiens, Advances in Theoreical and Applied Mahemaics, 6(1), pp. 41-46, (2011). 13. Elzaki, T.M. and Ezaki, S.M.Applicaions of new ransform Elzaki ransform o parial differenial equaions, Global Journal of Pure and Applied Mahemaics, 7(1), pp. 65-70, (2011). 14. Shendkar, A.M. and Jadhav, P.V.Elzaki ransform: A soluion of differenial equaions, Inernaional Journal of Science, Engineering and Technology Research, 4(4), pp. 1006-1008, 2015 51