Quadruple Simultaneous Fourier series Equations Involving Heat Polynomials
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- Αργυρός Μαγγίνας
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1 Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: Quruple Siulteous Fourier series Equtios Ivolvig Het Poloils Guj Shukl, K.C. Tripthi. Dr. Aekr Istitute of Teholog for Hippe, Kpur -8, Uttr Presh, Ii Defee Mterils & Stores Reserh & Developet Estlishet, Kpur-83, U.P. Ii Astrt: Quruple series equtios re useful i fiig the solutio of four prt our vlue proles of eletrosttis, elstiit other fiels of thetil phsis. I the preset pper, we hve osiere the quruple series equtios ivolvig het poloils solve the.. Itroutio Quruple series equtios re useful i fiig the solutio of four prt our vlue proles of eletrosttis, elstiit other fiels of thetil phsis. Cooke [] evise the etho for fiig the solutio of quruple series equtios ivolvig Fourier Bessel series otie their solutio usig opertor theor. Reetl Dwivei Trivei [] Dwivei Gupt [3] Dwivei Sigh [4] osiere vrious tpes of quruple series equtios ivolvig ifferet poloils. I the preset pper, we hve osiere the quruple series equtios ivolvig het poloils.. Quruple Siulteous Fourier Series Equtios Ivolvig Het Poloils Quruple series equtios ivolvig het poloils osiere here, re the geerliztio of ul series equtios osiere Pthk [3] orrespoig triple series equtios osiere. Solutio is otie reuig the prole to siulteous Frehol itegrl equtios of the seo ki. 3. The Equtios Here we shll osier the two sets of quruple series equtios ivolvig het poloils of the first ki seo ki respetivel. (i Quruple Series Equtios of the First Ki Quruple series equtios of the first ki to e stuie here re give s: A (. P p, σ(, t f (, t, < Γ µ (. t A P p, ν(, t f (, t, < < Γ ν A (.3 P p, σ(, t f 3(, t, < < Γ µ t A (.4 P p, ν(, t f 4(, t, < < Γ ν (ii Quruple Series Equtios of the Seo Ki Quruple series equtios of the seo ki to e lse, here re give s: t B (.5 P p, ν(, t g (, t, < Γ ν B (.6 P p, σ(, t g (, t, < < Γ µ t B P p, ν(, t g 3(, t, < < (.7 Γ ν B P p, σ(, t g 4(, t, < < Γ µ I ove equtios f i(, t g i(, t (i,, 3, 4 t > P (, t (.8 re the presrie futios, ν is the het poloils. B re the ukow oeffiiets to e eterie. A 4. The Solutio (i Equtios of the First Ki I orer to solve the quruple series equtios of the first ki, we set Volue 4 Issue, Jur 5 Pper ID: SUB Liese Uer Cretive Coos Attriutio CC BY
2 Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: φ (, t φ (, t re ukow futios. Usig reltio (. i equtios (., (.3, (. (., we oti Γ σ Γ µ A A f (, t φ(, t 4( p ( p! p Γ σ f 3(, t φ(, t W p, σ(, t Ω( (.3 Sustitutig this epressio for A i equtios (. (.4, p, ν t e Γ σ Γ µ P (, t 4( p ( p! Γ σ Γ ν f (, t φ (, t f 3(, t φ(, t W p, σ(, t Ω ( σ σ 3 F (, t f (, t f (, ts(,, t f (, ts(,, t 4 σ σ 3 (.3 (. F (, t f (, t f (, ts(,, t f (, ts(,, t Now usig the ottio give (.6 i equtio(., -σ 4t e φ (, t S ω(,, Γ( Γ( σ ν ν- 4t σ e φ(, t S (,, ν ω Γ( Γ( Puttig the vlue of Ω ( fro equtio (. hgig the orer of itegrtio sutio i ove equtios, σ σ σ σ f (, t (, t f 3(, t (, t φ φ σ p, ν p, σ t 4( p Γ µ p P (, tw (, t (.6 ( p! Γ σ Γ ν f (, t, < < f 4(, t, < < (.7 Usig the sutio result (.4 i equtios (.6 (.7, we hve σ σ f (, ts(,, t φ(, ts(,, t σ σ f 3(, ts(,, t φ (, ts(,, t σ σ φ (, ts(,, t φ (, ts(,, t Volue 4 Issue, Jur 5 F (, t, < < (.4 4t e φ(, ts (,, Γ( Γ( 4t 4t Pper ID: SUB Liese Uer Cretive Coos Attriutio CC BY e φ (, ts (,, e φ (, ts (,, ν F (, t, < < (.5 Puttig the vlue of sutio i ters of itegrl fro (.5 i equtio (.5, we oti ( e φ ( (, t η( νσ ( ( 4t νσ e φ (, t η( ( ( e φ (, t η( Γ( Γ( F (, t < < (.6 ν Chgig the orer of itegrtio i equtio (.6, η( e φ(, t νσ ( (
3 νσ ( ( Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: ( e (, t η φ η( e φ (, t νσ ( ( Γ( Γ( F (, t < < (.7 ν Assuig e (, t (.8 φ φ ( A ( φ ( e (, t (.9 φ ( Now i view of the equtio (4.8 equtio (4.7 e rewritte s η( φ Γ( Γ( F (, t νσ ν ( η( e φ (, t νσ ( ( η( e φ (, t < < νσ ( ( The solutio of the ove equtio is give s Si( νσ η( φ ( νσ ( ( ( (zz F (, t ν νσ Γ Γ νσ η z ( e φ (, t η(zz νσ ( z ( z e φ(, t, < < ( z Si( νσ η( φ ( F (, t νσ ( 3 η(zz e φ(, t νσ ( z ( z (. 4t η(zz e φ(, t νσ < < ( z ( z Si( νσ Γ( Γ( F (, t ν F (, t ( (. Chgig the orer of itegrtio i equtio (., Si( η( φ ( F 3(, t νσ η (zz e φ(, t νσ ( ( z ( z η(zz. z νσ ( ( νσ e φ (, t η(zz. ( z e φ (, t z νσ ( ( z ( z < < (. z νσ νσ ( ( z I equtio (4., Si( νσ η( φ { ( F 3(, t η(zz ( 4t ( z ( ( z (.3 Si( νσ z e φ (, t ( 4t ( z ( ( z z e φ (, t η(zz e φ(, t (zz η Si( νσ ( z < < (.4 Equtios (4.8 (4.9 re Ale tpe itegrl equtios, hee their solutios re give 4t Si(. φ( e φ(, t (.5 A ( ( Si(. φ( (.6 e φ(, t With the help of equtios (.5 (.6, we oti Volue 4 Issue, Jur 5 Pper ID: SUB Liese Uer Cretive Coos Attriutio CC BY
4 e (, t Si(. ( φ φ ( ( ( ( e φ (, t Si(. φ ( ( ( ( ( Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: (.7 (.8 Sustitutig the results (.7 (.8 is equtio (.4, we hve Si( νσ Si( η( φ ( F 3(, t ( νσ η(z ( z. φ( z ( z ( z ( (z( z ( z. ( z ( z ( z ( η(z ( z Si( φ( z ( z ( η φ < < (.9 Chgig the orer of itegrtio of the equtio (.9, Si( νσ Si( η( φ ( F 3(, t ( νσ. φ ( ( η(z z z ( ( z ( z, ( ( ( z ( z (z z z φ( η z ( ( ( z ( Si( φ (z ( η z < < (.3 Now, ove equtio e writte s 3 η( φ ( M(, φ ( F (, t N(, φ (, < < (.3 Si( νσsi( M(,. ( ( z, η(z ( z ( z ( z z (.3 Si( νσ Si( N(,. ( ( ( ( z ( z ( z η(z ( z ( ( z η(z z z Si( Agi strtig fro equtio (., we hve (.33 z σ σ φ φ (, ts(,, t (, ts(,, t F (, t, < < (.34 Usig the ottio give equtio (4.6, we oti σ 4t σ e φ (, t S w (,, ν Γ( Γ( σ 4t σ e φ(, t S w(,, ν Γ( Γ( F (, t, < < (.35 4t e φ(, ts (,, Γ( Γ( 4t 4t e (, ts (,, e (, ts (,, φ φ ν F (, t, < < (.36 Puttig the vlue of sutio i ters of itegrl fro (.5 i ove equtio, ( ( φ η e (, t ( ( ( ( ( e φ (, t η( e φ (, t η( Γ( Γ( F (, t < < (.37 ν Chgig the orer of itegrtio i equtio (.37, η( e φ(, t νσ ( ( Volue 4 Issue, Jur 5 Pper ID: SUB567 5 Liese Uer Cretive Coos Attriutio CC BY
5 νσ ( ( Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: ( e (, t η φ η( e φ(, t νσ ( ( η( e φ (, t νσ ( ( Γ( Γ( F (, t < < (.38 ν I view of the equtios (.8 (.9 ove equtio eoes η( φ( Γ( Γ( F (, t νσ ν ( η( e φ (, t νσ ( ( ( e (, t η φ νσ ( ( η( e φ (, t (.39 < < νσ ( ( The solutio to the equtio (.39 is give Si( νσ ( η( φ ( ( νσ ( ( (zz F (, t ν νσ Γ Γ νσ η e (, t (zz νσ φ η ( z ( z e φ (, t η(zz z νσ ( z ( z 4t e φ(, t < < (.4 ( z Let Si( νσ Γ( Γ( F 4 (, t ν F (, t (.4 νσ ( Usig (.4 i (.4 hgig the orer of itegrtio, Si( νσ { η( φ ( F 4(, t η (zz e φ (, t νσ ( ( z ( z η(zz. z νσ ( ( νσ e φ (, t z η(zz. ( z e (, t νσ φ, ( ( z ( z < < (.4 We kow tht ( ( νσ ( z ( ( z z νσ (.43 Usig the result (.43 i equtio (.4, Si( νσ { η φ 4 η ( 4t φ ( ( F (, t (zz z e (, t ( z ( ( z ( 4t (zz z ( z ( ( z ( 4t ( z ( ( z z e φ (, t η z e φ (, t η (zz, < < (.44 ( z ( Si( νσ η(z η( φ ( F 4(, t z ( z Volue 4 Issue, Jur 5 Pper ID: SUB567 5 Liese Uer Cretive Coos Attriutio CC BY
6 ( z Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: ( z ( z 4t e φ (, t η(z z z ( z ( z ( z 4t e φ (, t η(z z z 4t e φ(, t (.45, < < ( z Sustitutig the vlues fro (.7 (.8 i ove equtio, we oti Si( νσ Si( η( φ ( F 4(, t ( νσ η(z ( z ( z φ( z ( z ( z ( η(z ( z φ( z ( z ( z ( ( z ( Si( νσ η(z ( z 4t e φ (, t z, < < (.46 ( z Chgig the orer of itegrtio of equtio (.46, Si( νσ Si( η( φ ( F 4(, t ( νσ φ ( ( ( ( z ( z η(z z z ( νσ (z( z ( z ( z φ ( η z ( Si( νσ e φ(, t z ( ( z ( z ( z η(z (.47 z, < < Agi usig the vlue fro equtio (.7 i equtio (.47, Si( νσ Si( η( φ ( F 4(, t ( νσ φ ( ( ( ( z ( z η(z z z ( z z z ( ( ( z ( z φ ( η(z z z z z ( (z( z φ ( η z < < ( ( z ( z Equtio (.39 ow e writte s (.48 η( φ ( P(, φ ( F (, t Q(, φ (, 4 < < (.49 Where Si( νσ Si( P(,. ( ( νσ (z( z ( z ( z η z (.5 A η(z ( z ( z z ( z ( z η(z ( z ( z (.5 z z ( z ( z Equtios (.3 (.39 re Frehol itegrl equtios of the seo ki whih eterie φ φ (. ( Vlues of φ (, t φ (, t e eterie with the help of equtios (.5 (.6 respetivel. Fill, the oeffiiets A e opute fro equtio(.3, whih stisf the quruple series equtios ivolvig het poloils, of the first ki. ψ (, t ψ (, t re ukow futios. Usig equtio (. i equtios (.5, (.53, (. i equtios (.5, (.53, (.6 (.8, we oti Γ σ Γ µ B ψ(, t g (, t 4( p ( p! p Γ σ (, t g 4(, t (.54 ψ W p, σ(, t Ω( Sustitutig this epressio for B i equtios (.5 (.7, t e Γ σ Γ µ P p, ν(, t 4( p ( p! Γ σ Γ ν Volue 4 Issue, Jur 5 Pper ID: SUB567 5 Liese Uer Cretive Coos Attriutio CC BY
7 Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: ν G (, t, < (.66 ψ (, t g (, t ψ(, t g 4(, t Now W puttig p, σ(, t the Ω ( vlue of sutio i ters of itegrl fro (.5 i equtio (.66, we oti Usig the results (. i ove equtios hgig the orer of itegrtio sutio, σ σ σ σ (, t g (, t (, t g 4(, t ψ ψ p P p, (, t σ Γ µ ν t W p, σ(, t 4( p ( p! Γ σ Γ ν Usig sutio result (.4 i equtios (.57 (.58, we hve σ σ σ σ 4 ψ (, ts(,, t g (, ts(,, t ψ (, ts(,, t g (, ts(,, t σ σ ψ (, ts(,, t ψ (, ts(,, t σ G (, t g (, t g (, ts(,, t σ g 4(, ts(,, t (.63 e ψ(, t η( ψ ( ( Si( νσ G σ (, t g 3(, t g (, ts(,, t ( νσ Γ( Γ( σ g 4(, ts(,, t (.64 G (, t, < (.7 ν Strtig fro equtio (.6, usig ottio give Now equtio (.7 tkes the for (.6, e ψ(, t (.7 η( ψ σ 4t ( G 3(, t η( σ e, < ψ(, t S (,, ν ω ( Γ( Γ( Where Si( νσ Γ( Γ( σ 4t σ e G 3 (, t ψ(, t S (,, ν ω Γ( Γ( ν G (, t (.73 G (, t, < (.65 νσ ( 4t e ψ(, ts (,, Γ( Γ( Solvig the itegrl equtio (.69 s 4t Si( ψ( (.74 e ψ(, t 4t 4t e ψ(, ts (,, e ψ(, ts (,, ( With the help of equtio (.74, we oti Volue 4 Issue, Jur 5 Pper ID: SUB Liese Uer Cretive Coos Attriutio CC BY ( ( ( ( ( ( 4t νσ 4t νσ e ψ (, t η( e ψ (, t η( e ψ (, t η( Γ( Γ( G (, t, < (.67 ν Ivertig the orer of itegrtio of equtio (.67, η( e ψ (, t νσ ( ( νσ ( ( η( e ψ (, t Γ( Γ( G (, t, < (.68 Assuig, ν ψ ( e (, t ψ ( (.69 Now equtio (.68 e writte s η( e ψ(, t ( νσ ψ ( ( Γ( Γ( G (, t, < (.7 ν With the help of equtios (.8, we solve the ove equtio s
8 e ψ (, t Si( ψ ( ( ( ( ( Agi, let ψ ( e (, t (.76 ψ ( Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: (.75 Solvig (.76 siilrl s (.74 (.75 respetivel, 4t Si( ψ( e ψ(, t ( (.77 e ψ (, t Si( ψ( Now usig the equtio (.79 i (.7, Si( η( η( ψ ( G 3(, t ( ( ( ( ψ ( ( ( ( (.79 (.78 Equtio (.8 reues to the followig for η( ψ ( G 3(, t R(, ψ(, < (.8 Si( η( (.8 R(, ( ( ( Agi strtig fro equtio (.6 usig the ottio (.6, we hve σ 4t σ e ψ(, t S (,, ν ω Γ( Γ( σ 4t σ e ψ(, t S (,, ν ω Γ( Γ( G (, t, < < (.8 4t e ψ(, ts (,, Γ( Γ( 4t e ψ(, ts (,, 4t e ψ(, ts (,, ν G (, t, < < (.83 Puttig the vlue of sutio i ters of itegrl fro (.5 i ove equtio, ( ( ( ( ψ η e (, t ( e ψ (, t η( ( ( 4t νσ e ψ (, t η( Volue 4 Issue, Jur 5 Γ( Γ( G (, t, (.84 ν Ivertig the orer of itegrtio η( e ψ(, t νσ ( ( η ψ ( e (, t νσ ( ( Pper ID: SUB Liese Uer Cretive Coos Attriutio CC BY η( e ψ (, t νσ ( ( Γ( Γ( G (, t, < < (.85 ν Usig (.69 (.76 the ove equtio eoes η( ψ( Γ( Γ( G (, t νσ ν ( η( ψ ( η( νσ νσ ( ( ψ ( e (, t < < (.86 The equtio (.87 e solve s Si( νσ η( ψ ( ( νσ Γ( Γ( νσ η(z ψ(zz G (, t ν νσ z ( η(zz e ψ (, t, νσ ( z ( z < < (.87 Si( νσ η( ψ(g 4(, t ( νσ η(z ψ(z η(zz e ψ(, t z, νσ νσ ( z ( z ( z < < (.88 Where Si( νσ Γ( Γ( G 4 (, t ν G (, t (.89 νσ ( Si( νσ η( ψ ( G (, t ( νσ 4 z e ψ(, t G 3(z, t η(z νσ ( z ( z
9 4t η(zz e ψ(, t, νσ < < ( z ( z Itertiol Jourl of Siee Reserh (IJSR ISSN (Olie: Ie Coperius Vlue (3: 6.4 Ipt Ftor (3: (.9 Brekig the lst ter of the ove equtio ito two prts, Si( νσ η( ψ ( G 4(, t ( νσ G 3(z, tz η(zz e ψ(, t νσ νσ, ( z ( z ( z < < (.9 Chgig the orer of itegrtio, the equtio (.9 eoes Si( νσ η( ψ ( G 4(, t G 3(z, tz η (zz νσ ( ( z e ψ(, t νσ νσ ( ( z ( z < < (.9 We kow tht νσ ( ( z νσ ( z (.93 ( ( z Usig the result (.79 (.94 to the equtio (.93, Si( νσ η( ψ ( G 4(, t G 3(z, tz ( z ( z η (zz ( ( z ( ( z Si( ψ( < < (.94 ( z ( ( z Si( νσ G 3(z, t( z η( ψ ( G 4(, t z ( z ( ( Si( νσ Si( η(z( z z ( z( z ψ(, < < (.95 ( Now hgig the orer of itegrtio of the lst ter of the equtio (.96, Si( νσ G 3(z, t( z η( ψ ( G 4(, t z ( z ( Si( νσ Si( ψ( νσ ( ( (.96 Now equtio (.97 e rewritte s Si( νσ η( ψ ( S(, ψ ( G 4(, t νσ ( G 3(z, t( z, z< < (.97 ( z Where S(, is the setri kerel Si( νσ Si( νσ S(,. ( ( η(z( z z, (.98 ( z( z Equtios (.98 (.8 re Frehol itegrl equtios of the seo ki eterie ψ ( ψ (. Ψ (, t Ψ (, t e the opute fro equtios (.74 (.78 respetivel. Fill, the oeffiiets B e lulte with the help of equtio (.54 whih stisf the equtios fro (.5 to (.8. Prtiulr Cse If we let i equtio (. to (.8, the reue to the orrespoig triple series equtio ivolvig het poloils this solutio e show to gree with tht otie erlier for triple series equtios. Siilrl, we oti the orrespoig ul series equtios ivolvig het poloils. Akowlegeet Authors re thkful to Dr. G.K. Due Dr. A.P. Dwivei for their o-opertio & support provie to e for preprig the pper. Authors re lso thkful to Mr. Mish Ver, Sietist C DMSRDE, Kpur for his support. Referees [] Cooke, J.C. (99: The solutio of triple Quruple itegrl equtios Fourier Series Equtios,Qurt.J.Meh. Appl. Mth., 45, pp [] Dwivei, A.P. & Trivei, T.N. (97 : Quruple series equtios ivolvig Joi poloils, Pro. Nt. A. Si. Ii, 4 (A, pp [3] Dwivei, A.P. & Gupt, P. (98: Quruple series equtios ivolvig Joi Poloils of ifferet iies, At Cie Ii, 6 (M, pp [4] Dwivei, A.P. & Sigh, V.B. (997: Quruple series equtios ivolvig series of Joi poloils, I.J.Pure & Appl. Mth., 8, Volue 4 Issue, Jur 5 Pper ID: SUB Liese Uer Cretive Coos Attriutio CC BY
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