Hindwi Pulishing Corportion Astrct nd Applied Anlysis Volume 28, Article ID 395257, 2 pges doi:.55/28/395257 Reserch Article The Study of Triple Integrl Equtions with Generlized Legendre Functions B. M. Singh, J. Rokne, nd R. S. Dhliwl 2 Deprtment of Computer Science, University of Clgry, Clgry, AB, Cnd T2N-N4 2 Deprtment of Mthemtics nd Sttistics, University of Clgry, Clgry, AB, Cnd T2N-N4 Correspondence should e ddressed to R. S. Dhliwl, dhli.r@shw.c Received 28 April 28; Accepted 2 Septemer 28 Recommended y Lnce Littlejohn A method is developed for solutions of two sets of triple integrl equtions involving ssocited Legendre functions of imginry rguments. The solution of ech set of triple integrl equtions involving ssocited Legendre functions is reduced to Fredholm integrl eqution of the second kind which cn e solved numericlly. Copyright q 28 B. M. Singh et l. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited.. Introduction Dul integrl equtions involving Legendre functions hve een solved y Bloin. He pplied these equtions to prolems of potentil theory nd to torsion prolem. Lter on Pthk 2 nd Mndl 3 who considered dul integrl equtions involving generlized Legendre functions which hve more generl solution thn the ones considered y Bloin. Recently, Singh et l. 4 considered dul integrl equtions involving generlized Legendre functions, nd their results re more generl thn those in 3. In the nlysis of mixed oundry vlue prolems, we often encounter triple integrl equtions. Triple integrl equtions involving Legendre functions hve een studied y Srivstv 5. Triple integrl equtions involving Bessel functions hve lso een considered y Cooke 6 9, Trnter, Love nd Clements, Srivstv 2, nd most of these uthors reduced the solution into solution of Fredholm integrl eqution of the second kind. The relevnt references for dul nd triple integrl equtions re given in the ook of Sneddon 3. In this pper, method is developed for solutions of two sets of triple integrl equtions involving generlized Legendre functions in Sections 3 nd 4. Ech set of triple integrl equtions is reduced to Fredholm integrl eqution of the second kind which my e solved numericlly. The im of this pper is to find more generl solution for the type of
2 Astrct nd Applied Anlysis integrl equtions given in 5 nd to develop n esier method for solving triple integrl equtions in generl. 2. Integrl involved generlized Legendre functions nd some useful results We first summrize some known results needed in the pper. We find from 4, eqution 2, pge 33 tht ( ) 2 [ 2 μ ] μ sinh αc P μ ] cosh αc cos τx dτ c [ cosh αc cosh xc ] μ /2 H α x, 2. μ</2 ndfrom 4, weotin ) [ 2 ( 3/2 2 μ ] μ sinh αc ( ) ( ) 2 μ iτ c 2 μ iτ sinh fτ sin xτ P μ ] c cosh αc dτ c [ cosh xc cosh αc ] μ /2 H x α, 2.2 μ> /2 ndh denotes the Heviside unit function. Furthermore, c /f, f> nd P μ cosh αc is the generlized Legendre function defined in 5, pge 37.From /2 i τ/c 4, 6, the generlized Mehler-Fock trnsform is defined y ψ ( cosh αc ) P μ cosh αc ] F τ dτ, 2.3 nd its inversion formul is F τ fτ ( ) ( ) sinh fτ 2 2 μ iτ c 2 μ iτ c 2.4 [ ] ( ) cosh αc ψ cosh αc sinh αc dα. P μ /2 i τ/c Equtions 2. nd 2.2 re of form 2.3. From the inversion formul given y 2.4, 2., nd 2.2, it follows tht cos xτ τ sin xτ τ sinh fτ ( /2 μ i τ/c ) ( /2 μ i τ/c ) 2 /2 μ x [ sinh αc ] μp μ cosh αc ] dα [ cosh αc cosh xc ] μ /2, μ < 2, 2 /2 μ x 2.5 [ ] μp μ sinh αc cosh αc dα /2 i τ/c [ ] /2 μ, μ > cosh xc cosh αc 2. 2.6
B. M. Singh et l. 3 The inversion theorem for Fourier cosine trnsforms nd the results 2. nd 2.2 led to P μ /2 i τ/c P μ /2 i τ/c [ ] 2 cosh αc c sinhμ αc /2 μ α cos τs [ ] μ /2, μ < cosh αc cosh sc 2, 2.7 [ ] 2c cosh αc [ sinh μ αc /2 μ sin fτ ( /2 μ i τ/c ) ( /2 μ i τ/c ) sin τs [ ] /2 μ, μ > cosh sc cosh αc 2. α 2.8 If h t is monotoniclly incresing nd differentile for <t<nd h t / inthis intervl, then the solutions of the equtions t t f x dx [ ] α g t, < t <, <α<, 2.9 h t h x f x dx [ h x h t ] α g t, < t <, <α<, 2. re given y Sneddon 3 s f x sin α f x sin α d dx x d dx h t g t dt [ ] α, < x <, 2. h x h t x h t g t dt [ ] α, <x<, 2.2 h t h x respectively, the prime denotes the derivtive with respect to t. 3. Triple integrl equtions with generlized Legendre functions: set I In this section, we will find solution of the following triple integrl equtions: ( τa τ sinh τf 2 μ i τ ) ( c 2 μ i τ ) P μ ] c cosh αc dτ, <α<, 3. A τ P μ ] 2 cosh αc dτ f α, < α <, 3.2 ( τa τ sinh τf 2 μ 3 i τ ) ( c 2 μ 3 i τ ) P μ ] 3 c cosh αc dτ, < α <, 3.3
4 Astrct nd Applied Anlysis A τ is n unknown function to e determined, f α is known function, nd P μ cosh αc is the generlized Legendre function defined in Section 2 nd /2 < /2 i τ/c μ < /2, /2 <μ 2 < /2, μ 3 > /2. The tril solution of 3., 3.2, nd 3.3 cn e written s A τ ψ t cos τt dt, 3.4 ψ t is n unknown function to e determined. On integrting 3.4 y prts, we get A τ ψ sin τ τ τ ψ t sin τt dt, 3.5 the prime denotes the derivtive with respect to t. Sustituting 3.5 into 3.3, interchnging the order of integrtions nd using 2.2, we find tht 3.3 is stisfied identiclly. Sustituting 3.5 into 3. nd using the integrl defined y 2.2, weotin ψ [ ] /2 μ cosh c cosh αc α ψ t dt [ ] /2 μ, <α<. 3.6 cosh tc cosh αc Eqution 3.6 is equivlent to the following integrl eqution: d dα α c sinh tc ψ t dt [ ] /2 μ, <α<. 3.7 cosh tc cosh αc By sustituting 3.4 into 3.2, interchnging the order of integrtions nd using the integrl defined y 2. we find tht α ψ t dt c [ ] /2 μ2 cosh αc cosh tc ( ) 2 [ 2 μ ] μ2 2 sinh αc f α, < α <, μ 2 < 2. 3.8 For otining the solution of the prolem, we need to solve two Ael s type integrl equtions 3.7 nd 3.8. We ssume tht d c sinh tc ψ t dt dα [ ] /2 μ φ α, <α<. 3.9 cosh tc cosh αc α
B. M. Singh et l. 5 The ove eqution is of the sme form s 3.7 nd defined in different region. Eqution 3.9 is of form 2.2. Hence, the solution of the integrl eqution 3.9 cn e written s ψ t cos ( μ ) t φ α dα [ ] /2 μ, cosh cα cosh tc 2 <μ <,<t<. 3. 2 to The solution of Ael s type integrl equtions 2. together with 3.7 nd 3.9 le ψ t cos ( μ ) φ α dα [ ] /2 μ, cosh cα cosh tc 2 <μ <, <t<. 3. 2 Equtions 3. nd 3. men tht 3.7 is stisfied identiclly. Eqution 3.8 cn e rewritten in the form ψ t dt [ ] /2 μ2 cosh αc cosh tc c α 2 ( μ 2 ) f α [ sinh αc ] μ2, ψ t dt [ ] /2 μ2 cosh αc cosh tc <α<. 3.2 Sustituting the expression for ψ t from 3. nd 3. into the first nd second integrl of 3.2 we otin α S t dt [ ] /2 μ2 cosh αc cosh tc dt φ u dt F α [ ] /2 μ2 [ ] /2 μ2, <t<, cosh αc cosh tc cosh cu cosh tc 3.3 S t t φ u dt [ cosh cu cosh tc ] /2 μ, 3.4 F α 2 ( μ 2 ) f α [ sinh αc ] μ2 c cos ( μ ). 3.5 Assuming tht the right-hnd side of 3.3 is known function of α it hs the form of 2.9, whose solution is given y S t cos ( ) μ 2 d dt t c sinh cα F α dα [ ] /2 μ2 I t, <t<, cosh ct cosh cα 2 <μ 2 < 2, 3.6
6 Astrct nd Applied Anlysis I t cos ( ) μ 2 d dt t c sinh cα dα dp [ ] /2 μ2 ( ) /2 μ2 cosh ct cosh cα cosh cα cosh cp φ u du ( ) /2 μ2, < t <, cosh cu cosh cp 2 <μ 2 < 2. 3.7 From the integrl d t dt c sinh cα dα [ ] /2 μ2 [ ] /2 μ2 cosh ct cosh cα cosh cα cosh cp [ ] /2 μ2 c sinh ct cosh c cosh cp [ ] [ ] cosh ct cosh cp /2 μ2, p<<t, cosh ct cosh c 2 <μ 2 < 2, 3.8 we then otin c cos ( μ 2 ) sinh ct I t [ cosh ct cosh c ] /2 μ 2 φ u du [ ] /2 μ. cosh cu cosh cp ( cosh c cosh cp ) /2 μ2 dp [ cosh ct cosh cp ] 3.9 Eqution 3.4 is n Ael-type eqution. Hence, its solution is φ u cos ( μ ) d du u c sinh cv S v dv [ ] /2 μ, < u <, cosh vc cosh uc 2 <μ < 2, 3.2 R p φ u du [ ] /2 μ. 3.2 cosh cu cosh cp Sustituting the expression for φ u from 3.2 into 3.2, integrting y prts, nd finlly interchnging the order of integrtions in second integrl, we rrive t R p c cos ( μ ) [ S v sinh cv dv [ ] /2 μ [ ] /2 μ cosh c cosh cp cosh cv cosh c ( ) 2 μ S v sinh cv dv 3.22 v c sinh cu du [ ] 3/2 μ [ ] /2 μ ]. cosh cu cosh cp cosh cv cosh cu
B. M. Singh et l. 7 The integrl v c sinh cu du [ ] 3/2 μ [ ] /2 μ cosh cu cosh cp cosh cv cosh cu cosh cv cosh c /2 μ μ /2 cosh cv cosh cp cosh c cosh cp 3.23 p<<v, 2 <μ < 2 together with 3.22 le to R p c cos ( ) μ ( ) /2 μ cosh c cosh pc S ν sinh cν dν [ ][ ] /2 μ. cosh νc cosh pc cosh νc cosh c 3.24 From 3.9, 3.2, nd 3.24, weotin I t S ν K ν, t dν, 3.25 c 2 cos ( ) ( ) μ cos μ2 sinh ct sinh cν K ν, t 2[ cosh ct cosh c ] /2 μ 2 [ ] /2 μ cosh cν cosh c ( ) μ μ cosh c cosh cp 2 dp [ ][ ]. cosh ct cosh cp cosh cν cosh cp 3.26 From 3.25, 3.6 cn e written s S t S ν K ν, t dν cos ( ) μ 2 d dt t c sinh cα F α dα [ ] /2 μ2, < t <. 3.27 cosh ct cosh cα Eqution 3.27 is Fredholm integrl eqution of the second kind with kernel K ν, t. The kernel is defined y 3.26. The integrl in 3.26 cnnot e solved nlyticlly, ut for prticulr vlues of μ nd μ 2 the vlues of K ν, t cn e found numericlly. Hence, the numericl solution of Fredholm integrl eqution 3.27 cn e otined for prticulr vlue of f α, μ,ndμ 2 to find numericl vlues of S t. Mking use of 3.2, 3.,nd 3.,the numericl results for ψ t cn e otined. Finlly, mking use of 3.4 the numericl results for A τ cn e otined.
8 Astrct nd Applied Anlysis 4. Triple integrl equtions with generlized Legendre functions: set II In this section, we will find the solution of the following triple integrl equtions: τa τ P μ ] cosh αc dτ, <α<, 4. ( sinh τf 2 μ 2 i τ ) ( c 2 μ 2 i τ ) A τ P μ ] 2 c cosh αc dτ f α, <α<, 4.2 τa τ P μ 3 cosh αc ] dτ, < α, 4.3 μ > /2, /2 <μ 2 < /2, /2 <μ 3 < /2. We ssume tht τa τ P μ 3 cosh αc ] dτ M α, <α<. 4.4 The inversion formul for generlized Mehler-Fock trnsforms 2.4 together with 4.3 nd 4.4 implies tht A τ f ( sinh fτ 2 2 μ 3 i τ ) ( c 2 μ 3 i τ ) c 4.5 sinh uc P μ [ ] 3 /2 i τ/c cosh uc M u du. Multiplying 4. y sinh αc μ / cosh xc cosh αc /2 μ, integrting oth sides fromtox nd with respect to α, nd then using 2.6 we otin A τ sin xτ dτ, <x<. 4.6 Sustituting the vlue of A τ from 4.5 into 4.6, interchnging the order of integrtions, nd using the integrl 2.2, weget x sinh uc M u du [ ] /2 μ3, μ 3 >, <x<. 4.7 cosh xc cosh uc 2 Sustituting the vlue of A τ from 4.5 into 4.2 nd interchnging the order of integrtions we rrive t sinh uc M u K 2 u, α du f α, <α<, 4.8
B. M. Singh et l. 9 K 2 u, α ( f 2 ) ( 2 μ 2 i τ ) ( c 2 μ 3 i τ ) ( c 2 μ 3 i τ ) c 2 μ 2 i τ c 4.9 sinh 2 fτ P μ [ ] 3 μ /2 i τ/c cosh uc P 2 [ ] /2 i τ/c cosh αc dτ, nd then 2.8 nd 2.2 imply tht c K 2 u, α ( ) ( )[ ] μ2 [ ] μ3 /2 μ 2 /2 μ3 sinh αc sinh uc mx α,u [ ] /2 μ2 [ ] /2 μ3, cosh sc cosh αc cosh sc cosh uc 4. μ 3 > 2, μ 2 > 2. Eqution 4.7 is n Ael-type eqution nd hs the form 2.9. Hence, the solution of 4.7 is M u, <u<. 4. Using 4. nd 2.5, 4.8 cn e written in the form [ sinh uc ] μ3 M u du mx α,u [ ] /2 μ2 [ ] /2 μ3 cosh sc cosh αc cosh sc cosh uc ( ) ( )[ ] μ3 /2 μ 2 /2 μ2 sinh αc f α F α, c sy, <α<. 4.2 Using the formul du mx α,u α s du du, 4.3 we cn write 4.2 in the form α S s [ ] /2 μ2 F α cosh sc cosh αc [ cosh sc cosh αc ] /2 μ2 M u [ sinh uc ] μ 3 du [ ] /2 μ2, <α<, cosh sc cosh uc 4.4
Astrct nd Applied Anlysis S s s M u [ sinh uc ] μ 3 du [ ] /2 μ2, < s <. 4.5 cosh sc cosh αc Assuming tht the right-hnd side of 4.4 is known function eqution nd 4.4 hs the form of 2., hence the solution of 4.4 cn e written s S s c cos ( ) d F α sinh αc dα μ 2 [ ] /2 μ2 I s, <s<, s cosh αc cosh sc 2 <μ 2 < 2, 4.6 I s c cos ( ) d sinh αc dα μ 2 [ ] /2 μ2 cosh αc cosh sc s dp M u [ sinh cu ] μ 3 dα [ ] /2 μ2 [ ] /2 μ3, < s <. cosh pc cosh αc cosh pc cosh uc 4.7 Eqution 4.7 is simplified to I s c cos ( ) μ 2 sinh sc [ ] /2 μ2 cosh c cosh sc M u [ sinh cu ] μ 3 du [ ] /2 μ3, <s<. cosh cp cosh cu [ cosh cp cosh c ] /2 μ2 dp [ cosh sc cosh cp ] 4.8 Let R p M u [ sinh cu ] μ 3 du [ ] /2 μ3. 4.9 cosh pc cosh uc Eqution 4.5 is of the form of 2.9. Hence, its solution is M u [ sinh cu ] μ 3 c cos ( ) μ 3 d du u S s sinh sc [ ] /2 μ3, < u <. 4.2 cosh uc cosh sc
B. M. Singh et l. Sustituting the expression for M u from 4.2 into 4.9 nd integrting y prts nd then using the following integrl: s c sinh uc du ( ) 3/2 μ3 ( ) /2 μ3 cosh pc cosh uc cosh uc cosh sc [ cosh c cosh cs ] /2 μ 3 ( /2 μ3 )[ cosh cs cosh cp ][ cosh cp cosh c ] /2 μ3, 4.2 s<<p, 2 <μ 2 < 2, we find tht R p c cos ( μ 3 ) ( ) /2 μ3 cosh cp cosh c S u sinh cu du [ ][ ] /2 μ3. cosh cp cosh uc cosh c cosh cu 4.22 Mking use of 4.8, 4.9,nd 4.22, wefindtht I s S u K 2 u, s du, 4.23 c 2 cos ( ) ( ) μ 2 cos μ3 sinh sc sinh uc K 2 u, s 2[ cosh c cosh sc ] /2 μ 2 [ ] /2 μ3 cosh c cosh cu [ ] μ2 μ cosh cp cosh c 3 dp [ ][ ]. cosh sc cosh cp cosh cp cosh cu 4.24 Using 4.7 nd 4.23, 4.6 cn e written in the form S s S u K 2 u, s du c cos ( ) d μ 2 s F α sinh αc dα [ ] /2 μ2, < s <. cosh αc cosh sc 4.25 Eqution 4.25 is Fredholm integrl eqution of the second kind with kernel defined y 4.24. The Fredholm integrl eqution 4.25 my e solved to find numericl vlues of S s for prticulr vlues of f α. And hence from 4.2 nd 4.5, the numericl vlues for A τ cn e otined for prticulr vlues of f α, μ 2,ndμ 3.
2 Astrct nd Applied Anlysis 5. Conclusions The solution of the two sets of triple integrl equtions involving generlized Legendre functions is reduced to the solution of Fredholm integrl equtions of the second kind which cn e solved numericlly. References A. A. Bloin, Solutions of certin dul integrl equtions, Prikldny Mtemtik i Mekhnik, vol. 28, no. 6, pp. 5 23, 964, English trnsltion in Journl of Applied Mthemtics nd Mechnics, vol. 28, no. 6, pp. 227 236, 964. 2 R. S. Pthk, On clss of dul integrl equtions, Koninklijke Nederlne Akdemie vn Wetenschppen, vol. 4, no. 4, pp. 49 5, 978. 3 B. N. Mndl, A note on dul integrl equtions involving ssocited Legendre function, Interntionl Journl of Mthemtics nd Mthemticl Sciences, vol. 5, no. 3, pp. 6 64, 992. 4 B. M. Singh, J. Rokne, nd R. S. Dhliwl, The study of dul integrl equtions with generlized Legendre functions, Journl of Mthemticl Anlysis nd Applictions, vol. 34, no. 2, pp. 725 733, 25. 5 K. N. Srivstv, On some triple integrl equtions involving Legendre functions of imginry rgument, Journl of Muln Azd College of Technology, vol., pp. 54 67, 968. 6 J. C. Cooke, Triple integrl equtions, The Qurterly Journl of Mechnics nd Applied Mthemtics, vol. 6, no. 2, pp. 93 23, 963. 7 J. C. Cooke, Some further triple integrl eqution solutions, Proceedings of the Edinurgh Mthemticl Society, vol. 3, pp. 33 36, 963. 8 J. C. Cooke, The solution of triple integrl equtions in opertionl form, The Qurterly Journl of Mechnics nd Applied Mthemtics, vol. 8, no., pp. 57 72, 965. 9 J. C. Cooke, The solution of triple nd qudruple integrl equtions nd Fourier-Bessel series, The Qurterly Journl of Mechnics nd Applied Mthemtics, vol. 25, no. 2, pp. 247 263, 972. C. J. Trnter, Some triple integrl equtions, Proceedings of the Glsgow Mthemticl Assocition, vol. 4, pp. 2 23, 96. E. R. Love nd D. L. Clements, A trnsformtion of Cooke s tretment of some triple integrl equtions, Journl of the Austrlin Mthemticl Society. Series B, vol. 9, no. 3, pp. 259 288, 976. 2 N. Srivstv, On triple integrl equtions involving Bessel function s kernel, Journl of Muln Azd College of Technology, vol. 2, pp. 39 5, 988. 3 I. N. Sneddon, Mixed Boundry Vlue Prolems in Potentil Theory, North-Hollnd, Amsterdm, The Netherln, 966. 4 A. Erdélyi, W. Mgnus, F. Oerhettinger, nd F. G. Tricomi, Tles of Integrl Trnsforms. Vol. II,McGrw Hill, New York, NY, USA, 954. 5 A. Erdélyi, W. Mgnus, F. Oerhettinger, nd F. G. Tricomi, Tles of Integrl Trnsforms. Vol. I,McGrw Hill, New York, NY, USA, 954. 6 W. Mgnus, F. Oerhettinger, nd R. P. Soni, Formuls nd Theorems for the Specil Functions of Mthemticl Physics, Die Grundlehren der Mthemtischen Wissenschften. 52, Springer, New York, NY, USA, 966.