SlideShare a Scribd company logo

H - FUNCTION AND GENERAL CLASS OF POLYNOMIAL AND HEAT CONDUCTION IN A ROD.

International Journal of Technical Research & Application, profile picture
International Journal of Technical Research & Application

This paper evaluates finite integrals involving general classes of polynomials and h-functions, applying these to a boundary value problem concerning heat conduction in a rod. The research establishes an expansion formula related to h-functions and demonstrates how specific coefficients can reduce the results to previously known cases. Keywords include general polynomials and h-functions, among others.

Engineering
1 of 5
Download to read offline
1
2
3
4
5
International Journal of Technical Research and Applications e-ISSN: 2320-8163, www.ijtra.com Volume 2, Issue 4 (july-aug 2014), PP. 122-126 122 | P a g e H- FUNCTION AND GENERAL CLASS OF POLYNOMIAL AND HEAT CONDUCTION IN A ROD. Dr. Rachna Bhargava Department of Mathematics Global College of Technology (GCT), Jaipur-302022, India Abstract - In this paper, first we evaluate a finite integral involving general class of polynomials and the product of two H -functions and then we make its application to solve boundary value problem on heat conduction in a rod under the certain conditions and further we establish an expansion formula involving about product of H -function. In view of generality of the polynomials and products of H -function occurring here in, on specializing the coefficients of polynomials and parameters of the H -function, our results would readily reduce to a large number of results involving known class of polynomials and simpler functions. Keywords: General Class of Polynomials, H Function, Jacobi polynomial and Leguerre polynomials. Mathematics Subject Classification : 33C60, 34B05 I. INTRODUCTION The general class of polynomials introduced by Shrivastava [7] and defined by [8] and [10] as follows: 0,1,2,...nx !k An) x]S kkn,km, [n/m] 0k m n      ….. (1) where m is an arbitrary positive integer the coefficient An,k (n,k ≥ 0) are arbitrary constants, real or complex. H -function will be defined and represented as follows [2] and [4]:          P1,NjjaN1,jAjj(a Q1,MjBjjbM1,jjb NM, QP, NM, QP, zHz]H        dz i2 1 i i …(2) where  ≠ 0 and        jj P 1Nj jB jj Q 1Mj jA jj N 1j jj M 1j ab1 a1b …(3) and also the H -function occurring in the paper was introduced by Inayat-Hussain [4] and studied by Bushman and Shrivastava [2]. The following series representation for the H -function was obtained by Rathie [5].          P1,NjjcN1,jCjj(c Q1,MjDjjdM1,jjd NM, QP, NM, QP, zHz]H rh, h r rh,jj P 1j jD rh,jj Q 1Mj jC rh,jj N 1j rh,jj M hj 1jM 1h0r z !r 1 cd1 c1d                 r r rh, hd    …(4) The nature of contour L and series of various conditions on its parameters can be seen in the paper by Bushman and Shrivastava [2]. We shall also make use of the following behaviour of the z]H NM, QP,  function for small value of f(z) as recorded by Saxena [6, p.112, eq.(2.3) and (2.4)]   |z|(0z}H NM, QP, for small z where (2)for)/(dRemin jj Mj1    and (4)for)/(bRemin jj Mj1    The following more general conditions given by 0TT 2 1 (zarg 111  and 0TT 2 1 (zarg 222  . where 0BAT j 1P 11Nj jj 1Q 11Mj jj 1N 1j j 1M 1j 1    and 0DCT j 2P 12Nj jj 2Q 12Mj jj 2N 1j j 2M 1j 2    .
123 | P a g e II. MAIN INTEGRAL                                        2b 2 2m 2n 1b 1 1m 1n m 1uL 0 L x sinyS L x sinyS L x sin L x sin                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH dx L x sinzH 2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P                    1k 1 1 1k1n1k1m1rh,2h2k2b1k1b2M 12h0r 2m2n 02k 1m1[n 01k y k An 2k             2 siny k An m2k 2 2 2k2n2k2m2     !r z1 cd1 c1d h rh, 2 r rh,jj 2P 12Nj jD rh,jj 2Q 12Mj jC rh,jj 2N 1j rh,jj 2M hj 1j                        1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,1MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  …(5) Where (i) 1u 2Lk   (ii) 0kandknnhh 212121  and provided that conditions (i) h h j j rb and0 b Remin             (ii) h h rh, j j rd and0 d Remin             (iii) 0TBA 1j 1P 11Nj jj 1Q 11Mj jj 1N 1j j 1M 1j    where  11 T 2 1 zarg (iv) 0qDC 2j 2P 12Nj jj 2Q 12Mj jj 2N 1j j 2M 1j    where  22 T 2 1 zarg (v) 0hhkbkb{uRe 1rh,22211  Proof : To establish the above integral (5), we first express both the general class of polynomials and z]H 2N2M 2Q2p    occurring in its left hand side in their respective series forms with the help of equation (1) and (3) respectively and then interchange the order of integration and summation (which is permissible under the condition stated) and using (3) and with the help of x-integral given by Gradshteyn, I.S. and Ryzhik [3]. Then we substitute the above  with the help of (4) and reinterpret the result thus in terms of H -function, we arrive at the right hand side of desired results (5). III. MAIN PROBLEM Problem of heat conduction in a rod with one end held at zero temperature and the other end exchanges heat freely with the surrounding medium at zero temperature. If the thermal coefficients are constants and there are no source of thermal energy, then temperature in a one- dimensional rod 0  x  L satisfies the following heat equation 0t t k t 2 2       …(6) In view of the problem, the solution of this partial differential equation satisfy the boundary conditions 0t)0  …(7) 0t)L,ht)L, t    …(8)  (x,t) is finite as t   …(9) The initial condition  (x,0) = f(x) …(10) The solution of partial differential equation (6) can be written as [11, p.77,(4)]                         kt 2 L m mm 1m e L x sinBt)x, …(11) at t = 0                                        2b 2 2m 2n 1b 1 1m 1n 1u L x sinyS L x sinyS L x sinf(x)x,0)                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH …(12) The solution of the problem to be obtained is          rh,2b2k2b1k1b2m 12h0r 2m2n 02k 1m1[n 01k 2k't)x,
124 | P a g e 2k 2 2 2k2n2k2m2 1k 1 1 1k1n1k1m1 y k An y !k An    !r z)1( )c()}d1({ )}c1({)d( h 2 r rh,jj P 1Nj D rh,jj Q 1Mj C rh,jj N 1j rh,jj M hj 1j rh, 2 2 j 2 2 j 22                                                  kt L exp L x sin 2sin2 2 sin 2 m m mm m m             1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  …(13) Where L2 d 4 k' 1u    valid under the condition of (5). Proof: The solution of the problem stated is given as [11, p.77, (4)]                         kt 2 L m mm 1m e L x sinBt)x, where 1, 2,… are rots of transcendental equation. kL tan m m   If t = 0, then by virtue of (11) and (12), we have                                       2b 2 2m 2n 1b 1 1m 1n 1u L x sinyS L x sinyS L x sin                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH L x sinB m m 1m      …(13) Multiplying both sides of (13) by L x sin m   and integrate with respect to x from 0 to L, we get                                        2b 2 2m 2n 1b 1 1m 1nm 1uL 0 L x sinyS L x sinyS L x sin L x sin                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH dx L x sin L x sinB mm L 0 m 1m        …(14) and using (5) and orthogonality property [12, p.28] by Szego, we obtain mm m 21 m m 2sin2 2 sin d 4 B                              rh,2b2k2b1k1b2m 12h0r 2m2n 02k 1m1[n 01k 2k' 2 siny k An y !k An m2k 2 2 2k2n2k2m2 1k 1 1 1k1n1k1m1     !r z1 cd1 c1d h rh, 2 r rh,jj 2P 12Nj jD rh,jj 2Q 12Mj jC rh,jj 2N 1j rh,jj 2M hj 1j                                 1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  …(15) with the help of (14) and (11), we arrive at the right hand side of desired result. IV. EXPANSION FORMULA                                       2b 2 2m 2n 1b 1 1m 1n 1u L x sinyS L x sinyS L x sin                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH          rh,2b2k2b1k1b2m 12h0r 2m2n 02k 1m1[n 01k 2k' 2k 2 2 2k2n2k2m2 1k 1 1 1k1n1k1m1 y k An y !k An    !r z1 cd1 c1d h rh, 2 r rh,jj 2P 12Nj jD rh,jj 2Q 12Mj jC rh,jj 2N 1j rh,jj 2M hj 1j           
125 | P a g e                    L x sin 2 sin mm m                      1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  …(16) where all the conditions of (5) are satisfied. Proof: Using (12) and (15) in (11), we arrive at the expansion formula 14 2L d 4 k'     6. SPECIAL CASES OF (12) (i) If k kn, 1 1)n n n A           then 2x)1Px]s ( n ' n   where x)P ( n   is Jacobi polynomial [10, p.68, eq. (4.3.2)] and also k kn, 1 1 n n A1,m         then x)Lx]S ( Ln ' n   where  Ln L (x) is Leguerre Polynomial [10, p.101, eq. (5.1.6)] and we get                         L x sinL L x sin21P L x sinf(x) 2n1n 1u                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjd 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH          rh,2b2k2b1k1b2M 12h0r 2m2n 02k 1m1[n 01k 2k' . 2k 2 2 2k2m2 1k 1 1 1k1m1 y k n y !k n                                         kt L exp L x sin 2sin2 2 sin 2 mm mm m m !r z1 cd1 c1d h rh, 2 r rh,jj 2P 12Nj jD rh,jj 2Q 12Mj jC rh,jj 2N 1j rh,jj 2M hj 1j                                         1 1 n n 1 1n n n 2 2 1k 1k1 1 1                      1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  (ii) If we substitute  1k(0,A1k(0,A0nn0mm 212121 1N1M 1Q1Pjj2222221 H,1BAand0h1QM0PN0nn    then it reduces to Fox H-function and Q)1,...,j;P1,...,i(Cjj  it reduces to the well known Meijer’s G-function by [9] in (12) then we get a known result given in [1]. On applying the same procedure as above in (16) , then we can establish the other known results. ACKNOWLEDGEMENT The authors are thankful to refree for his valuable suggestions. REFERENCES 1. Bajpai, S.D. and Mishra, Sadhana; (1991) Meijer’s G-function and Fox’s H-function and Heat conduction in a rod under typical boundary conditions, Jnanabha, Vol.21 . 2. Bushman, R.G. and Shrivastava, H.M. (1990) The H -function associated with a certain class of Feynman integral. J. Phys. A: Math. Gen. 23, 4707-4710 . 3. Gradshteyn, I.S. and Ryzhik, I.M. (1980) Tables of Integrals, Series and Products, Academic Press, Inc. New York . 4. Inayat-Hussain,,A.A.(1987) New Properties of Hypergeometric series derivable from Feynman integrals: II A generalization of the H-function, J. Phys. A: Math. Gen. 20 , 4119- 4128. 5. Rathie,A.K.(1997) A new generalization of generalized hypergeometric function, Le Matematiche Fasc II, 52 , 297-310. 6. Saxena, R.K., Chena, Ram and Kalla, S.L. (2002) Application of generalized H-function in vibariate distribution, Rev. Acad. Can. Ci.enc. XIV (Nums 1-2) , 111-120. 7. Shrivastava, H.M. (1972) A contour integral involving Fox’s H-function, Indian J. Math. 14 , 1-6. 8. Shrivastava, H.M. (1983) The Weyl fractional integral of a general class of polynomials, Bull. Un. Math. Ital. (6) 2B , 219-228.
126 | P a g e 9. Shrivastava, H.M., Gupta, K.C. and Goyal, S.P. (1982) The H-Function of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras . 10. Shrivastava, H.M. and Singh, N.P. (1983) The integration of certain products of the multivariable H-function with a general class of polynomials, Rend. Circ. Mat. Palermo 2(32) , 157-187. 11. Sommerfeld, A. (1949) Partial Differential Equations in Physics, Academic Press, New York . 12. Szego, G. (1975) Orthogonal polynomials. (Amer. Math. Soc. Collog. Publ. Vol.23), 4th ed., Amer. Math. Soc. Providence, Rhode Island .

More Related Content

PDF
The Queue Length of a GI M 1 Queue with Set Up Period and Bernoulli Working V...
YogeshIJTSRD
 
PDF
-type and -type four dimensional plane wave solutions of Einstein's field eq...
inventy
 
PDF
A new non symmetric information divergence of
eSAT Publishing House
 
PDF
Second-order Cosmological Perturbations Engendered by Point-like Masses
Maxim Eingorn
 
PDF
Duel of cosmological screening lengths
Maxim Eingorn
 
PDF
Compensating Joint Configuration through Null Space Control in Composite Weig...
Waqas Tariq
 
PDF
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...
cscpconf
 
PDF
International Journal of Mathematics and Statistics Invention (IJMSI)
inventionjournals
 
The Queue Length of a GI M 1 Queue with Set Up Period and Bernoulli Working V...
YogeshIJTSRD
 
-type and -type four dimensional plane wave solutions of Einstein's field eq...
inventy
 
A new non symmetric information divergence of
eSAT Publishing House
 
Second-order Cosmological Perturbations Engendered by Point-like Masses
Maxim Eingorn
 
Duel of cosmological screening lengths
Maxim Eingorn
 
Compensating Joint Configuration through Null Space Control in Composite Weig...
Waqas Tariq
 
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...
cscpconf
 
International Journal of Mathematics and Statistics Invention (IJMSI)
inventionjournals
 

What's hot (19)

PDF
Effect of Michell’s Function in Stress Analysis Due to Axisymmetric Heat Supp...
IJERA Editor
 
PDF
Fixed Point Theorem in Fuzzy Metric Space
IJERA Editor
 
DOCX
Dealinggreensfncsolft sqrd(10 5-2k16)
foxtrot jp R
 
DOCX
Sweeping discussions on dirac field1 update3 sqrd
foxtrot jp R
 
DOCX
Dealinggreensfncsolft
foxtrot jp R
 
PDF
Radiation effects on heat and mass transfer of a mhd
eSAT Publishing House
 
PDF
An Approach to Analyze Non-linear Dynamics of Mass Transport during Manufactu...
BRNSS Publication Hub
 
PDF
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERS
Zac Darcy
 
PDF
Various relations on new information
ijitjournal
 
PDF
Sweeping discussion on_dirac_fields_update1
foxtrot jp R
 
PDF
ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS ...
ijoejournal
 
PDF
On Approach to Increase Integration Rate of Elements of an Operational Amplif...
BRNSS Publication Hub
 
PDF
Uniformity of the Local Convergence of Chord Method for Generalized Equations
IOSR Journals
 
DOCX
Dealinggreensfncsolft sqrdb
foxtrot jp R
 
DOCX
Sw2gr1 sqrd
foxtrot jp R
 
PDF
Crack problems concerning boundaries of convex lens like forms
ijtsrd
 
PDF
MODELING OF REDISTRIBUTION OF INFUSED DOPANT IN A MULTILAYER STRUCTURE DOPANT...
mathsjournal
 
PDF
Stability Analysis for Steady State Solutions of Huxley Equation
Mohammad Sabawi Lecturer at Mathematics Department/College of Educations for Women/Tikrit University
 
PDF
Quantum Annealing for Dirichlet Process Mixture Models with Applications to N...
Shu Tanaka
 
Effect of Michell’s Function in Stress Analysis Due to Axisymmetric Heat Supp...
IJERA Editor
 
Fixed Point Theorem in Fuzzy Metric Space
IJERA Editor
 
Dealinggreensfncsolft sqrd(10 5-2k16)
foxtrot jp R
 
Sweeping discussions on dirac field1 update3 sqrd
foxtrot jp R
 
Dealinggreensfncsolft
foxtrot jp R
 
Radiation effects on heat and mass transfer of a mhd
eSAT Publishing House
 
An Approach to Analyze Non-linear Dynamics of Mass Transport during Manufactu...
BRNSS Publication Hub
 
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERS
Zac Darcy
 
Various relations on new information
ijitjournal
 
Sweeping discussion on_dirac_fields_update1
foxtrot jp R
 
ANALYSIS OF MANUFACTURING OF VOLTAGE RESTORE TO INCREASE DENSITY OF ELEMENTS ...
ijoejournal
 
On Approach to Increase Integration Rate of Elements of an Operational Amplif...
BRNSS Publication Hub
 
Uniformity of the Local Convergence of Chord Method for Generalized Equations
IOSR Journals
 
Dealinggreensfncsolft sqrdb
foxtrot jp R
 
Sw2gr1 sqrd
foxtrot jp R
 
Crack problems concerning boundaries of convex lens like forms
ijtsrd
 
MODELING OF REDISTRIBUTION OF INFUSED DOPANT IN A MULTILAYER STRUCTURE DOPANT...
mathsjournal
 
Stability Analysis for Steady State Solutions of Huxley Equation
Mohammad Sabawi Lecturer at Mathematics Department/College of Educations for Women/Tikrit University
 
Quantum Annealing for Dirichlet Process Mixture Models with Applications to N...
Shu Tanaka
 
Ad

Viewers also liked (20)

PDF
CASE OF IDENTIFICATION OF SNEAKERS BY TRACE EVIDENCE ANALYSIS OF THE REMAININ...
International Journal of Technical Research & Application
 
PDF
Waste_Management_System_-_FYP_3
Tarrall Yanzu
 
DOCX
Bus 370 entire course
profanchloren1984
 
PDF
EQUILIBRIUM, KINETIC AND THERMODYNAMIC STUDIES ON BASIC DYE ADSORPTION USING ...
International Journal of Technical Research & Application
 
PDF
Weplan
Alicia Zunzunegui García
 
PDF
About TrueNorth Consulting
Richard B Stephenson
 
DOC
Resume
Ravi Kant Mishra
 
PDF
Untitled Presentation
Joyeria Stylos
 
PDF
El óvalo imperfecto
ncalavia
 
PPTX
Как организовать приём промо-кодов через сайт и смс
rusloterei
 
PDF
THE INFLUENCE OF ETHICAL IDEOLOGIES ON ATTITUDES TOWARD SUICIDE
International Journal of Technical Research & Application
 
PDF
PERFORMANCE COMPARISON OF AODV AND DSDV ROUTING PROTOCOLS IN WIRELESS AD HOC ...
International Journal of Technical Research & Application
 
PDF
DEVELOPMENT OF A MULTIAGENT BASED METHODOLOGY FOR COMPLEX SYSTEMS
International Journal of Technical Research & Application
 
PDF
first tatoo
Shailja Jha
 
DOC
Sufiyan_CV_Upated- DEC- 2014
Mohammed Sufiyan
 
PPTX
Using powerpoint at exhibitions presentation
Lorna Rose
 
DOCX
Preliminary task evaluation grid
chloeread
 
PDF
BIO-CHEMICAL CHANGES OF NUTRIENTS IN RICE PLANT UNDER SODIC SOILS
International Journal of Technical Research & Application
 
DOC
ACTUAL_TR_
Murat Akinci
 
PDF
GeoEng12334
Edward Antonio
 
CASE OF IDENTIFICATION OF SNEAKERS BY TRACE EVIDENCE ANALYSIS OF THE REMAININ...
International Journal of Technical Research & Application
 
Waste_Management_System_-_FYP_3
Tarrall Yanzu
 
Bus 370 entire course
profanchloren1984
 
EQUILIBRIUM, KINETIC AND THERMODYNAMIC STUDIES ON BASIC DYE ADSORPTION USING ...
International Journal of Technical Research & Application
 
Weplan
Alicia Zunzunegui García
 
About TrueNorth Consulting
Richard B Stephenson
 
Resume
Ravi Kant Mishra
 
Untitled Presentation
Joyeria Stylos
 
El óvalo imperfecto
ncalavia
 
Как организовать приём промо-кодов через сайт и смс
rusloterei
 
THE INFLUENCE OF ETHICAL IDEOLOGIES ON ATTITUDES TOWARD SUICIDE
International Journal of Technical Research & Application
 
PERFORMANCE COMPARISON OF AODV AND DSDV ROUTING PROTOCOLS IN WIRELESS AD HOC ...
International Journal of Technical Research & Application
 
DEVELOPMENT OF A MULTIAGENT BASED METHODOLOGY FOR COMPLEX SYSTEMS
International Journal of Technical Research & Application
 
first tatoo
Shailja Jha
 
Sufiyan_CV_Upated- DEC- 2014
Mohammed Sufiyan
 
Using powerpoint at exhibitions presentation
Lorna Rose
 
Preliminary task evaluation grid
chloeread
 
BIO-CHEMICAL CHANGES OF NUTRIENTS IN RICE PLANT UNDER SODIC SOILS
International Journal of Technical Research & Application
 
ACTUAL_TR_
Murat Akinci
 
GeoEng12334
Edward Antonio
 
Ad

Similar to H - FUNCTION AND GENERAL CLASS OF POLYNOMIAL AND HEAT CONDUCTION IN A ROD. (20)

PPT
Btech admission in india
Edhole.com
 
PDF
Boundary value problem and its application in i function of multivariable
Alexander Decker
 
PDF
Convolution of an Fractional integral equation with the I-function as its kernel
IRJET Journal
 
PPT
M220w07
Aafaq Malik
 
PPTX
23MA401 Numerical Methods Boundary value problems
shyamalaseec
 
PDF
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
ijceronline
 
PDF
History of the Dirichlet Problem for Laplace's equation.pdf
hutong1
 
PPT
Btech admission in india
Edhole.com
 
PDF
A Boundary Value Problem and Expansion Formula of I - Function and General Cl...
IJERA Editor
 
PDF
Green’s Function Solution of Non-homogenous Singular Sturm-Liouville Problem
IJSRED
 
PDF
On Some Integrals of Products of H -Functions
IJMER
 
PDF
Application Of The Theory Of Linear Singular Integral Equations And Contour I...
Cheryl Brown
 
PDF
Sub1567
International Journal of Science and Research (IJSR)
 
DOCX
Digital Text Book :POTENTIAL THEORY AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Baasilroy
 
PDF
H function and a problem related to a string
Alexander Decker
 
PDF
D0621619
IOSR Journals
 
PDF
I- Function and H -Function Associated With Double Integral
IOSR Journals
 
PDF
Heat problems
Ho Linh
 
PDF
Formulas for Surface Weighted Numbers on Graph
ijtsrd
 
DOCX
mathematics question bank for engineering students
MrMRubanVelsUniversi
 
Btech admission in india
Edhole.com
 
Boundary value problem and its application in i function of multivariable
Alexander Decker
 
Convolution of an Fractional integral equation with the I-function as its kernel
IRJET Journal
 
M220w07
Aafaq Malik
 
23MA401 Numerical Methods Boundary value problems
shyamalaseec
 
IJCER (www.ijceronline.com) International Journal of computational Engineerin...
ijceronline
 
History of the Dirichlet Problem for Laplace's equation.pdf
hutong1
 
Btech admission in india
Edhole.com
 
A Boundary Value Problem and Expansion Formula of I - Function and General Cl...
IJERA Editor
 
Green’s Function Solution of Non-homogenous Singular Sturm-Liouville Problem
IJSRED
 
On Some Integrals of Products of H -Functions
IJMER
 
Application Of The Theory Of Linear Singular Integral Equations And Contour I...
Cheryl Brown
 
Sub1567
International Journal of Science and Research (IJSR)
 
Digital Text Book :POTENTIAL THEORY AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Baasilroy
 
H function and a problem related to a string
Alexander Decker
 
D0621619
IOSR Journals
 
I- Function and H -Function Associated With Double Integral
IOSR Journals
 
Heat problems
Ho Linh
 
Formulas for Surface Weighted Numbers on Graph
ijtsrd
 
mathematics question bank for engineering students
MrMRubanVelsUniversi
 

More from International Journal of Technical Research & Application (20)

PDF
STUDY & PERFORMANCE OF METAL ON METAL HIP IMPLANTS: A REVIEW
International Journal of Technical Research & Application
 
PDF
EXPONENTIAL SMOOTHING OF POSTPONEMENT RATES IN OPERATION THEATRES OF ADVANCED...
International Journal of Technical Research & Application
 
PDF
POSTPONEMENT OF SCHEDULED GENERAL SURGERIES IN A TERTIARY CARE HOSPITAL - A T...
International Journal of Technical Research & Application
 
PDF
STUDY OF NANO-SYSTEMS FOR COMPUTER SIMULATIONS
International Journal of Technical Research & Application
 
PDF
ENERGY GAP INVESTIGATION AND CHARACTERIZATION OF KESTERITE CU2ZNSNS4 THIN FIL...
International Journal of Technical Research & Application
 
PDF
POD-PWM BASED CAPACITOR CLAMPED MULTILEVEL INVERTER
International Journal of Technical Research & Application
 
PDF
DIGITAL COMPRESSING OF A BPCM SIGNAL ACCORDING TO BARKER CODE USING FPGA
International Journal of Technical Research & Application
 
PDF
MODELLING THE IMPACT OF FLOODING USING GEOGRAPHIC INFORMATION SYSTEM AND REMO...
International Journal of Technical Research & Application
 
PDF
AN EXPERIMENTAL STUDY ON SEPARATION OF WATER FROM THE ATMOSPHERIC AIR
International Journal of Technical Research & Application
 
PDF
LI-ION BATTERY TESTING FROM MANUFACTURING TO OPERATION PROCESS
International Journal of Technical Research & Application
 
PDF
QUALITATIVE RISK ASSESSMENT AND MITIGATION MEASURES FOR REAL ESTATE PROJECTS ...
International Journal of Technical Research & Application
 
PDF
SCOPE OF REPLACING FINE AGGREGATE WITH COPPER SLAG IN CONCRETE- A REVIEW
International Journal of Technical Research & Application
 
PDF
IMPLEMENTATION OF METHODS FOR TRANSACTION IN SECURE ONLINE BANKING
International Journal of Technical Research & Application
 
PDF
EFFECT OF TRANS-SEPTAL SUTURE TECHNIQUE VERSUS NASAL PACKING AFTER SEPTOPLASTY
International Journal of Technical Research & Application
 
PDF
EVALUATION OF DRAINAGE WATER QUALITY FOR IRRIGATION BY INTEGRATION BETWEEN IR...
International Journal of Technical Research & Application
 
PDF
THE CONSTRUCTION PROCEDURE AND ADVANTAGE OF THE RAIL CABLE-LIFTING CONSTRUCTI...
International Journal of Technical Research & Application
 
PDF
TIME EFFICIENT BAYLIS-HILLMAN REACTION ON STEROIDAL NUCLEUS OF WITHAFERIN-A T...
International Journal of Technical Research & Application
 
PDF
A STUDY ON THE FRESH PROPERTIES OF SCC WITH FLY ASH
International Journal of Technical Research & Application
 
PDF
AN INSIDE LOOK IN THE ELECTRICAL STRUCTURE OF THE BATTERY MANAGEMENT SYSTEM T...
International Journal of Technical Research & Application
 
PDF
OPEN LOOP ANALYSIS OF CASCADED HBRIDGE MULTILEVEL INVERTER USING PDPWM FOR PH...
International Journal of Technical Research & Application
 
STUDY & PERFORMANCE OF METAL ON METAL HIP IMPLANTS: A REVIEW
International Journal of Technical Research & Application
 
EXPONENTIAL SMOOTHING OF POSTPONEMENT RATES IN OPERATION THEATRES OF ADVANCED...
International Journal of Technical Research & Application
 
POSTPONEMENT OF SCHEDULED GENERAL SURGERIES IN A TERTIARY CARE HOSPITAL - A T...
International Journal of Technical Research & Application
 
STUDY OF NANO-SYSTEMS FOR COMPUTER SIMULATIONS
International Journal of Technical Research & Application
 
ENERGY GAP INVESTIGATION AND CHARACTERIZATION OF KESTERITE CU2ZNSNS4 THIN FIL...
International Journal of Technical Research & Application
 
POD-PWM BASED CAPACITOR CLAMPED MULTILEVEL INVERTER
International Journal of Technical Research & Application
 
DIGITAL COMPRESSING OF A BPCM SIGNAL ACCORDING TO BARKER CODE USING FPGA
International Journal of Technical Research & Application
 
MODELLING THE IMPACT OF FLOODING USING GEOGRAPHIC INFORMATION SYSTEM AND REMO...
International Journal of Technical Research & Application
 
AN EXPERIMENTAL STUDY ON SEPARATION OF WATER FROM THE ATMOSPHERIC AIR
International Journal of Technical Research & Application
 
LI-ION BATTERY TESTING FROM MANUFACTURING TO OPERATION PROCESS
International Journal of Technical Research & Application
 
QUALITATIVE RISK ASSESSMENT AND MITIGATION MEASURES FOR REAL ESTATE PROJECTS ...
International Journal of Technical Research & Application
 
SCOPE OF REPLACING FINE AGGREGATE WITH COPPER SLAG IN CONCRETE- A REVIEW
International Journal of Technical Research & Application
 
IMPLEMENTATION OF METHODS FOR TRANSACTION IN SECURE ONLINE BANKING
International Journal of Technical Research & Application
 
EFFECT OF TRANS-SEPTAL SUTURE TECHNIQUE VERSUS NASAL PACKING AFTER SEPTOPLASTY
International Journal of Technical Research & Application
 
EVALUATION OF DRAINAGE WATER QUALITY FOR IRRIGATION BY INTEGRATION BETWEEN IR...
International Journal of Technical Research & Application
 
THE CONSTRUCTION PROCEDURE AND ADVANTAGE OF THE RAIL CABLE-LIFTING CONSTRUCTI...
International Journal of Technical Research & Application
 
TIME EFFICIENT BAYLIS-HILLMAN REACTION ON STEROIDAL NUCLEUS OF WITHAFERIN-A T...
International Journal of Technical Research & Application
 
A STUDY ON THE FRESH PROPERTIES OF SCC WITH FLY ASH
International Journal of Technical Research & Application
 
AN INSIDE LOOK IN THE ELECTRICAL STRUCTURE OF THE BATTERY MANAGEMENT SYSTEM T...
International Journal of Technical Research & Application
 
OPEN LOOP ANALYSIS OF CASCADED HBRIDGE MULTILEVEL INVERTER USING PDPWM FOR PH...
International Journal of Technical Research & Application
 

Recently uploaded (20)

PDF
A SYSTEMATIC REVIEW OF APPLICATIONS IN FRAUD DETECTION
IJNSA Journal
 
DOCX
ASol_English-Language-Literature-Set-1-27-02-2023-converted.docx
AYUSHMANAV
 
PDF
Enhancing Cyber Defense Against Zero-Day Attacks using Ensemble Neural Networks
IJCNCJournal
 
PDF
Level 2 – IBM Data and AI Fundamentals (1)_v1.1.PDF
KSRIHARISASANKA
 
PDF
keyrequirementskkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
mojeeburahmanwardak
 
PDF
BIO-INSPIRED HORMONAL MODULATION AND ADAPTIVE ORCHESTRATION IN S-AI-GPT
ijaia
 
PPTX
Fundamentals of Mechanical Engineering.pptx
MdMowdudAhmed
 
PPTX
Engineering Ethics, Safety and Environment [Autosaved] (1).pptx
MehrabTaseenTalukder
 
PDF
R24 SURVEYING LAB MANUAL for civil enggi
MNANDITHACIVILSTAFF
 
PPTX
UNIT 4 Total Quality Management .pptx
gokuld13012005
 
PDF
PPT on Performance Review to get promotions
prashant593122
 
PDF
Automation-in-Manufacturing-Chapter-Introduction.pdf
pcmth867
 
PPTX
M Tech Sem 1 Civil Engineering Environmental Sciences.pptx
ShriNRPrasad
 
PPTX
Construction Project Organization Group 2.pptx
vj5agdales
 
PPTX
UNIT-1 - COAL BASED THERMAL POWER PLANTS
Chandra Kumar S
 
PPT
Mechanical Engineering MATERIALS Selection
arsalanahmad705384
 
PPT
Introduction, IoT Design Methodology, Case Study on IoT System for Weather Mo...
MariaRufina4
 
PDF
Mohammad Mahdi Farshadian CV - Prospective PhD Student 2026
Educational Group Mohammad Farshadian
 
PDF
The CXO Playbook 2025 – Future-Ready Strategies for C-Suite Leaders Cerebrai...
Cerebraix Technologies
 
PPTX
Foundation to blockchain - A guide to Blockchain Tech
Davies Chacko
 
A SYSTEMATIC REVIEW OF APPLICATIONS IN FRAUD DETECTION
IJNSA Journal
 
ASol_English-Language-Literature-Set-1-27-02-2023-converted.docx
AYUSHMANAV
 
Enhancing Cyber Defense Against Zero-Day Attacks using Ensemble Neural Networks
IJCNCJournal
 
Level 2 – IBM Data and AI Fundamentals (1)_v1.1.PDF
KSRIHARISASANKA
 
keyrequirementskkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk
mojeeburahmanwardak
 
BIO-INSPIRED HORMONAL MODULATION AND ADAPTIVE ORCHESTRATION IN S-AI-GPT
ijaia
 
Fundamentals of Mechanical Engineering.pptx
MdMowdudAhmed
 
Engineering Ethics, Safety and Environment [Autosaved] (1).pptx
MehrabTaseenTalukder
 
R24 SURVEYING LAB MANUAL for civil enggi
MNANDITHACIVILSTAFF
 
UNIT 4 Total Quality Management .pptx
gokuld13012005
 
PPT on Performance Review to get promotions
prashant593122
 
Automation-in-Manufacturing-Chapter-Introduction.pdf
pcmth867
 
M Tech Sem 1 Civil Engineering Environmental Sciences.pptx
ShriNRPrasad
 
Construction Project Organization Group 2.pptx
vj5agdales
 
UNIT-1 - COAL BASED THERMAL POWER PLANTS
Chandra Kumar S
 
Mechanical Engineering MATERIALS Selection
arsalanahmad705384
 
Introduction, IoT Design Methodology, Case Study on IoT System for Weather Mo...
MariaRufina4
 
Mohammad Mahdi Farshadian CV - Prospective PhD Student 2026
Educational Group Mohammad Farshadian
 
The CXO Playbook 2025 – Future-Ready Strategies for C-Suite Leaders Cerebrai...
Cerebraix Technologies
 
Foundation to blockchain - A guide to Blockchain Tech
Davies Chacko
 

H - FUNCTION AND GENERAL CLASS OF POLYNOMIAL AND HEAT CONDUCTION IN A ROD.

  • 1. International Journal of Technical Research and Applications e-ISSN: 2320-8163, www.ijtra.com Volume 2, Issue 4 (july-aug 2014), PP. 122-126 122 | P a g e H- FUNCTION AND GENERAL CLASS OF POLYNOMIAL AND HEAT CONDUCTION IN A ROD. Dr. Rachna Bhargava Department of Mathematics Global College of Technology (GCT), Jaipur-302022, India Abstract - In this paper, first we evaluate a finite integral involving general class of polynomials and the product of two H -functions and then we make its application to solve boundary value problem on heat conduction in a rod under the certain conditions and further we establish an expansion formula involving about product of H -function. In view of generality of the polynomials and products of H -function occurring here in, on specializing the coefficients of polynomials and parameters of the H -function, our results would readily reduce to a large number of results involving known class of polynomials and simpler functions. Keywords: General Class of Polynomials, H Function, Jacobi polynomial and Leguerre polynomials. Mathematics Subject Classification : 33C60, 34B05 I. INTRODUCTION The general class of polynomials introduced by Shrivastava [7] and defined by [8] and [10] as follows: 0,1,2,...nx !k An) x]S kkn,km, [n/m] 0k m n      ….. (1) where m is an arbitrary positive integer the coefficient An,k (n,k ≥ 0) are arbitrary constants, real or complex. H -function will be defined and represented as follows [2] and [4]:          P1,NjjaN1,jAjj(a Q1,MjBjjbM1,jjb NM, QP, NM, QP, zHz]H        dz i2 1 i i …(2) where  ≠ 0 and        jj P 1Nj jB jj Q 1Mj jA jj N 1j jj M 1j ab1 a1b …(3) and also the H -function occurring in the paper was introduced by Inayat-Hussain [4] and studied by Bushman and Shrivastava [2]. The following series representation for the H -function was obtained by Rathie [5].          P1,NjjcN1,jCjj(c Q1,MjDjjdM1,jjd NM, QP, NM, QP, zHz]H rh, h r rh,jj P 1j jD rh,jj Q 1Mj jC rh,jj N 1j rh,jj M hj 1jM 1h0r z !r 1 cd1 c1d                 r r rh, hd    …(4) The nature of contour L and series of various conditions on its parameters can be seen in the paper by Bushman and Shrivastava [2]. We shall also make use of the following behaviour of the z]H NM, QP,  function for small value of f(z) as recorded by Saxena [6, p.112, eq.(2.3) and (2.4)]   |z|(0z}H NM, QP, for small z where (2)for)/(dRemin jj Mj1    and (4)for)/(bRemin jj Mj1    The following more general conditions given by 0TT 2 1 (zarg 111  and 0TT 2 1 (zarg 222  . where 0BAT j 1P 11Nj jj 1Q 11Mj jj 1N 1j j 1M 1j 1    and 0DCT j 2P 12Nj jj 2Q 12Mj jj 2N 1j j 2M 1j 2    .
  • 2. 123 | P a g e II. MAIN INTEGRAL                                        2b 2 2m 2n 1b 1 1m 1n m 1uL 0 L x sinyS L x sinyS L x sin L x sin                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH dx L x sinzH 2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P                    1k 1 1 1k1n1k1m1rh,2h2k2b1k1b2M 12h0r 2m2n 02k 1m1[n 01k y k An 2k             2 siny k An m2k 2 2 2k2n2k2m2     !r z1 cd1 c1d h rh, 2 r rh,jj 2P 12Nj jD rh,jj 2Q 12Mj jC rh,jj 2N 1j rh,jj 2M hj 1j                        1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,1MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  …(5) Where (i) 1u 2Lk   (ii) 0kandknnhh 212121  and provided that conditions (i) h h j j rb and0 b Remin             (ii) h h rh, j j rd and0 d Remin             (iii) 0TBA 1j 1P 11Nj jj 1Q 11Mj jj 1N 1j j 1M 1j    where  11 T 2 1 zarg (iv) 0qDC 2j 2P 12Nj jj 2Q 12Mj jj 2N 1j j 2M 1j    where  22 T 2 1 zarg (v) 0hhkbkb{uRe 1rh,22211  Proof : To establish the above integral (5), we first express both the general class of polynomials and z]H 2N2M 2Q2p    occurring in its left hand side in their respective series forms with the help of equation (1) and (3) respectively and then interchange the order of integration and summation (which is permissible under the condition stated) and using (3) and with the help of x-integral given by Gradshteyn, I.S. and Ryzhik [3]. Then we substitute the above  with the help of (4) and reinterpret the result thus in terms of H -function, we arrive at the right hand side of desired results (5). III. MAIN PROBLEM Problem of heat conduction in a rod with one end held at zero temperature and the other end exchanges heat freely with the surrounding medium at zero temperature. If the thermal coefficients are constants and there are no source of thermal energy, then temperature in a one- dimensional rod 0  x  L satisfies the following heat equation 0t t k t 2 2       …(6) In view of the problem, the solution of this partial differential equation satisfy the boundary conditions 0t)0  …(7) 0t)L,ht)L, t    …(8)  (x,t) is finite as t   …(9) The initial condition  (x,0) = f(x) …(10) The solution of partial differential equation (6) can be written as [11, p.77,(4)]                         kt 2 L m mm 1m e L x sinBt)x, …(11) at t = 0                                        2b 2 2m 2n 1b 1 1m 1n 1u L x sinyS L x sinyS L x sinf(x)x,0)                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH …(12) The solution of the problem to be obtained is          rh,2b2k2b1k1b2m 12h0r 2m2n 02k 1m1[n 01k 2k't)x,
  • 3. 124 | P a g e 2k 2 2 2k2n2k2m2 1k 1 1 1k1n1k1m1 y k An y !k An    !r z)1( )c()}d1({ )}c1({)d( h 2 r rh,jj P 1Nj D rh,jj Q 1Mj C rh,jj N 1j rh,jj M hj 1j rh, 2 2 j 2 2 j 22                                                  kt L exp L x sin 2sin2 2 sin 2 m m mm m m             1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  …(13) Where L2 d 4 k' 1u    valid under the condition of (5). Proof: The solution of the problem stated is given as [11, p.77, (4)]                         kt 2 L m mm 1m e L x sinBt)x, where 1, 2,… are rots of transcendental equation. kL tan m m   If t = 0, then by virtue of (11) and (12), we have                                       2b 2 2m 2n 1b 1 1m 1n 1u L x sinyS L x sinyS L x sin                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH L x sinB m m 1m      …(13) Multiplying both sides of (13) by L x sin m   and integrate with respect to x from 0 to L, we get                                        2b 2 2m 2n 1b 1 1m 1nm 1uL 0 L x sinyS L x sinyS L x sin L x sin                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH dx L x sin L x sinB mm L 0 m 1m        …(14) and using (5) and orthogonality property [12, p.28] by Szego, we obtain mm m 21 m m 2sin2 2 sin d 4 B                              rh,2b2k2b1k1b2m 12h0r 2m2n 02k 1m1[n 01k 2k' 2 siny k An y !k An m2k 2 2 2k2n2k2m2 1k 1 1 1k1n1k1m1     !r z1 cd1 c1d h rh, 2 r rh,jj 2P 12Nj jD rh,jj 2Q 12Mj jC rh,jj 2N 1j rh,jj 2M hj 1j                                 1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  …(15) with the help of (14) and (11), we arrive at the right hand side of desired result. IV. EXPANSION FORMULA                                       2b 2 2m 2n 1b 1 1m 1n 1u L x sinyS L x sinyS L x sin                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjc 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH          rh,2b2k2b1k1b2m 12h0r 2m2n 02k 1m1[n 01k 2k' 2k 2 2 2k2n2k2m2 1k 1 1 1k1n1k1m1 y k An y !k An    !r z1 cd1 c1d h rh, 2 r rh,jj 2P 12Nj jD rh,jj 2Q 12Mj jC rh,jj 2N 1j rh,jj 2M hj 1j           
  • 4. 125 | P a g e                    L x sin 2 sin mm m                      1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  …(16) where all the conditions of (5) are satisfied. Proof: Using (12) and (15) in (11), we arrive at the expansion formula 14 2L d 4 k'     6. SPECIAL CASES OF (12) (i) If k kn, 1 1)n n n A           then 2x)1Px]s ( n ' n   where x)P ( n   is Jacobi polynomial [10, p.68, eq. (4.3.2)] and also k kn, 1 1 n n A1,m         then x)Lx]S ( Ln ' n   where  Ln L (x) is Leguerre Polynomial [10, p.101, eq. (5.1.6)] and we get                         L x sinL L x sin21P L x sinf(x) 2n1n 1u                    1P11Njja 1N1,jAjja 1Q11MjBjjb 1M1,jj(b 1h 1 1N1M 1Q1P L x sinzH                    2P12Njjd 2N1,jCjjc 2Q12MjDjjd 2M1,jj(d 2h 2 2N2M 2Q2P L x sinzH          rh,2b2k2b1k1b2M 12h0r 2m2n 02k 1m1[n 01k 2k' . 2k 2 2 2k2m2 1k 1 1 1k1m1 y k n y !k n                                         kt L exp L x sin 2sin2 2 sin 2 mm mm m m !r z1 cd1 c1d h rh, 2 r rh,jj 2P 12Nj jD rh,jj 2Q 12Mj jC rh,jj 2N 1j rh,jj 2M hj 1j                                         1 1 n n 1 1n n n 2 2 1k 1k1 1 1                      1P11Njja 1N1,jAjja11h1rh,2h2k2b1k1bu 11hmrh,2h2k2b1k1bu 1Q1,MjBjjb 1M1,jj(bh 111N1M 11Q11P 2 z H  (ii) If we substitute  1k(0,A1k(0,A0nn0mm 212121 1N1M 1Q1Pjj2222221 H,1BAand0h1QM0PN0nn    then it reduces to Fox H-function and Q)1,...,j;P1,...,i(Cjj  it reduces to the well known Meijer’s G-function by [9] in (12) then we get a known result given in [1]. On applying the same procedure as above in (16) , then we can establish the other known results. ACKNOWLEDGEMENT The authors are thankful to refree for his valuable suggestions. REFERENCES 1. Bajpai, S.D. and Mishra, Sadhana; (1991) Meijer’s G-function and Fox’s H-function and Heat conduction in a rod under typical boundary conditions, Jnanabha, Vol.21 . 2. Bushman, R.G. and Shrivastava, H.M. (1990) The H -function associated with a certain class of Feynman integral. J. Phys. A: Math. Gen. 23, 4707-4710 . 3. Gradshteyn, I.S. and Ryzhik, I.M. (1980) Tables of Integrals, Series and Products, Academic Press, Inc. New York . 4. Inayat-Hussain,,A.A.(1987) New Properties of Hypergeometric series derivable from Feynman integrals: II A generalization of the H-function, J. Phys. A: Math. Gen. 20 , 4119- 4128. 5. Rathie,A.K.(1997) A new generalization of generalized hypergeometric function, Le Matematiche Fasc II, 52 , 297-310. 6. Saxena, R.K., Chena, Ram and Kalla, S.L. (2002) Application of generalized H-function in vibariate distribution, Rev. Acad. Can. Ci.enc. XIV (Nums 1-2) , 111-120. 7. Shrivastava, H.M. (1972) A contour integral involving Fox’s H-function, Indian J. Math. 14 , 1-6. 8. Shrivastava, H.M. (1983) The Weyl fractional integral of a general class of polynomials, Bull. Un. Math. Ital. (6) 2B , 219-228.
  • 5. 126 | P a g e 9. Shrivastava, H.M., Gupta, K.C. and Goyal, S.P. (1982) The H-Function of One and Two Variables with Applications, South Asian Publishers, New Delhi and Madras . 10. Shrivastava, H.M. and Singh, N.P. (1983) The integration of certain products of the multivariable H-function with a general class of polynomials, Rend. Circ. Mat. Palermo 2(32) , 157-187. 11. Sommerfeld, A. (1949) Partial Differential Equations in Physics, Academic Press, New York . 12. Szego, G. (1975) Orthogonal polynomials. (Amer. Math. Soc. Collog. Publ. Vol.23), 4th ed., Amer. Math. Soc. Providence, Rhode Island .