{"id":1737,"date":"2018-12-05T06:42:56","date_gmt":"2018-12-05T06:42:56","guid":{"rendered":"http:\/\/www.gyanvihar.org\/journals\/?p=1737"},"modified":"2019-05-22T10:55:01","modified_gmt":"2019-05-22T10:55:01","slug":"a-solution-of-partial-differential-equation-associated-with-i-function-and-generalized-m-series-in-the-study-of-angular-displacement","status":"publish","type":"post","link":"https:\/\/www.gyanvihar.org\/journals\/a-solution-of-partial-differential-equation-associated-with-i-function-and-generalized-m-series-in-the-study-of-angular-displacement\/","title":{"rendered":"A SOLUTION OF PARTIAL DIFFERENTIAL EQUATION ASSOCIATED WITH I- FUNCTION AND GENERALIZED M-SERIES IN THE STUDY OF ANGULAR DISPLACEMENT"},"content":{"rendered":"<p style=\"text-align: center;\">pp 34-37<br \/>\nHARSHITA GARG1, ASHOK SINGH SHEKHAWAT2<br \/>\n1Research Scholar, Suresh Gyan Vihar University, Jaipur.<br \/>\n2Professor,Arya College of Engineering and Information Technology, Jaipur.<\/p>\n<p>Abstract: In an attempt to give extension of the result in the theory of special functions, we discuss the application of certain products involving I- function 10 and a generalized M-series 12 in obtaining a solution of the partial differential equation\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1742\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/eq1-1.png\" alt=\"\" width=\"90\" height=\"50\" \/>Concerning to a problem of angular displacement in a shaft.<\/p>\n<p>Key-words: I-function, partial differential equation, generalized M-series, angular displacement.<\/p>\n<p>INTRODUCTION<br \/>\nThe I- function given by Saxena [10] is represented and defined as following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1745\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/eq2.png\" alt=\"\" width=\"447\" height=\"339\" \/><\/p>\n<p style=\"text-align: justify;\">pi (i= 1,\u2026.,r), qi (i= 1,\u2026.,r), m, n are integers satisfying 0 \u2264 n \u2264 pi, 0 \u2264 m \u2264 qi (i= 1,\u2026.,r); r is finite, \u03b1j, \u03b2j, \u03b1ji, \u03b2ji, are real and positive; aj, bj, aji, bji are complex numbers and \u00a3 is the path of integration separating the increasing and decreasing sequences of poles of the integrand. The integral converges if\u00a0\u00a0arg|(z) &lt; | ( \u03c0\/2\u2126i<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1747\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/e4.png\" alt=\"\" width=\"433\" height=\"249\" \/><\/p>\n<p>If we take r=1 in (1.1), then the I- function will convert to the well known Fox\u2019s H- function. The generalized M-series is the extension of the both Mitag- Laffler function and generalized hyper geometric function. It is represented as following:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1750\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/e5.png\" alt=\"\" width=\"495\" height=\"158\" \/><\/p>\n<p>Here (uj)k ,(vj)k are the known pochammer symbols. The series (1.4) is defined when none of the parameters vj\u2019s (j= 1, 2,\u2026, q), is a negative integer or zero. If any numerator parameter uj is a negative integer or zero then the series terminates to a polynomial in z. The series (1.4) is convergent for all y if p \u2264 q. We consider the problem of determining the twist f(y,t) in a shaft of circular section with its axis\u00a0along the y-axis. Now the displacement f(y,t) due to initial twist must satisfy the boundary value problem. If we assume that both the ends y = 0 and y = \u03c5 of the shaft are free<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1752\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/e6.png\" alt=\"\" width=\"344\" height=\"583\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1754\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/e8.png\" alt=\"\" width=\"341\" height=\"65\" \/><\/p>\n<p style=\"text-align: justify;\">Proof: The integral in (2.1) can be established by using the definition of generalizes M-series given by (1.4) and I-function in terms of Mellin-Barnes contour integral given by (1.1), then interchanging the order of summation and integration, obtain the inner integral with the help of a result given by\u00a0Chaurasia and Gupta [2], and we reach at the desired result.<br \/>\n2. Solution of the Problem posed: The solutionof the problem to be established is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1760\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/3-4.png\" alt=\"\" width=\"321\" height=\"153\" \/><\/p>\n<p>This is valid under the same conditions required for (2.1)<\/p>\n<p>3. Derivation of (3.1): The solution of the problem can be written as ([4], Churchill, 1941, p.125 (4)).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1762\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/4-3.png\" alt=\"\" width=\"294\" height=\"46\" \/><\/p>\n<p>Where a =\u03c4( 0,1,2,&#8230;)\u03c4 are the coefficients in the Fourier Cosine Series for \u03c8(y) in the interval (0, \u03c5). If t = 0, then by virtue of (1.7), we get<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1764\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/5-2.png\" alt=\"\" width=\"353\" height=\"325\" \/><\/p>\n<p>Now by using (2.1) along with orthogonal property of the cosine functions, we get<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1770\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/51.png\" alt=\"\" width=\"352\" height=\"506\" \/><\/p>\n<p>Valid under the conditions which are true for (2.1) and (5.1) (ii) Taking generalized polynomials [14] in place of M-series in (2.1), we get<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1772\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/6-2.png\" alt=\"\" width=\"258\" height=\"172\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1774\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/52.png\" alt=\"\" width=\"352\" height=\"145\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1780\" src=\"http:\/\/www.gyanvihar.org\/journals\/wp-content\/uploads\/2018\/12\/12-2.png\" alt=\"\" width=\"350\" height=\"115\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Now taking r = 1 and s = 2 and \u21920 \u03bbi in (5.1), we get the known result obtained by Chaurasia and Godika [1]. (ii) Taking i \u21921 and r \u21921 in (5.1), we get the known result obtained by Chaurasia and Shekhawat [3]<br \/>\n(iii) Taking Aleph function in place of I- function in (5.1), we get the known result obtained by Shekhawat and Garg (13).<\/p>\n<p style=\"text-align: justify;\">CONCLUSION<br \/>\nIn this paper, the established result is very useful in many interesting situations appearing in the literature on mathematical analysis, applied mathematics and mathematical physics with the help of our result. We found the angular<br \/>\ndisplacement in a shaft \u2013III by using special function (I- function).<\/p>\n<p style=\"text-align: justify;\">ACKNOWLEDGEMENT<br \/>\nThe authors are thankful to Professor H. M. Srivastava, University of Victoria, Canada for valuable suggestions, which have led the paper to this present form.<\/p>\n<p style=\"text-align: justify;\">REFERENCES<br \/>\n[1]. Chaurasia, V.B.L. and Godika, A., A solution ofthe partial differential equation of angular displacement in shaft \u2013 II, Acta Ciencia Indica, 23M (1), 77-89 (1997).<br \/>\n[2]. Chaurasia, V.B.L. and Gupta, V.G., The H function of several complex variables and angular displacement in a shaft \u2013 II, Indian J. Pure Appl. Math., 14(5), 588-595 (1983).<br \/>\n[3]. Chaurasia, V.B.L. and Shekhawat, A.S., An application of H -function and a generalized polynomials in the study of angular displacement in a shaft \u2013 II, Appl. Sci. Period 8, No.1, 37-46 (2006).<br \/>\n[4]. Churchill, R.V., Fourier series and boundary value problems, McGraw-Hill Book Co., Inc. New York (1941).<br \/>\n[5]. Dutta, B.K., Arora, L.K. and Borah, J., on the solution of fractional kinetic equation, Gen. Math. Notes, Vol.6, No.1, 40-48 (2011).<br \/>\n[6]. Fox, C., The G and H-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc.98, 395- 429 (1961).<br \/>\n[7]. Saxena, R.K., Mathai, A.M. and Haubold, H.J., On generalized fractional kinetic equations, Phys. A., 344, 653-664 (2004).<br \/>\n[8]. Saxena, R.K. and Pogany, T.K., on fractional integration formula for Aleph functions, Appl. Math. Comput., 218(3), 985-990 (2011).<br \/>\n[9]. Saxena, R.K. and Pogany, T.K.: Mathieu-type series for the Aleph-function occurring in Fokker-Planck equation, Eur. J. Pure Appl. Math., 3(6), 958-979 (2010).<br \/>\n[10]. Saxena, V.P., The I-function, Anamaya Publishers, New Delhi (2008).<br \/>\n[11]. Sharma, M.: Fractional Integration and Fractional Differentiation of the M-Series. J. Fract. Calc. and Appl. Anal. Vol. 11, No. 2, 187- 191 (2008).<br \/>\n[12]. Sharma, M. and Jain, R.: A note on a generalized M-Series as a special function of fractional calculus. J. Fract. Calc. and Appl. Anal. Vol. 12, No. 4, 449-452 (2009).<br \/>\n[13]. Shekhawat, A.S. and Garg, H.: Angular displacement in a shaft associated with the Aleph function and generalized polynomials. , Res. J. Mathematical and statistical Sci. Vol. 1(10), 1-4 (2013).<br \/>\n[14]. Srivastava, H.M., A multilinear generating\u00a0 function for the Konhauser sets of bi-orthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 117, 183-191 (1985).<br \/>\n[15]. S\u00fcdland, N., Baumann, B. and Nonnenmacher, T.F., Who knows about the Aleph (\u2135)-function? Fract. Calc. Appl. Anal., 1(4), 401-402, (1998).<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>pp 34-37 HARSHITA GARG1, ASHOK SINGH SHEKHAWAT2 1Research Scholar, Suresh Gyan Vihar University, Jaipur. 2Professor,Arya College of Engineering and Information Technology, Jaipur. Abstract: In an attempt to give extension of the result in the theory of special functions, we discuss the application of certain products involving I- function 10 and a generalized M-series 12 in [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[46],"tags":[],"class_list":["post-1737","post","type-post","status-publish","format-standard","hentry","category-volume-1-issue-1-2015-journal-of-engineering-and-technology"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>research journal - Research Journal<\/title>\n<meta name=\"description\" content=\"A Solution Of Partial Differential Equation Associated With I- Function And Generalized M-series In The Study Of Angular Displacement Journal | SGVU\" \/>\n<meta name=\"robots\" content=\"index, follow, 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