Semi-Analytical Solution of the 1D Helmholtz Equation, Obtained from Inversion of Symmetric Tridiagonal Matrix

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1 Journal of Electromagnetc nalyss and pplcatons, 04, 6, Publshed Onlne December 04 n ScRes. Sem-nalytcal Soluton of the D Helmholtz Equaton, Obtaned from Inverson of Symmetrc Trdagonal Matr Sergne Bra Gueye Département de Physque, Faculté des Scences et Technques, Unversté Chekh nta Dop, Dakar-Fann, Sénégal Emal: sbragy@gmal.com Receved October 04; revsed ovember 04; accepted 5 ovember 04 Copyrght 04 by author and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons ttrbuton Internatonal Lcense (CC BY). bstract n nterestng sem-analytc soluton s gven for the Helmholtz equaton. Ths soluton s obtaned from a rgorous dscusson of the regularty and the nverson of the trdagonal symmetrc matr. Then, applcatons are gven, showng very good accuracy. Ths work provdes also the analytcal nverse of the skew-symmetrc trdagonal matr. Keywords Helmholtz Equaton, Trdagonal Matr, Lnear Homogeneous Recurrence Relaton. Introducton We focus on the nverse of the matr (M) defned n the Equaton () below. We are nterested n applcatons of ths matr, because the latter allows solvng many mportant dfferental equatons n scence and technology, especally mathematcs, physcs, engneerng, chemstry, bology and other dscplnes. The formula of the nverse of (M) was determned n []. But n ths study, a dfferent approach s presented wth a rgourus and complete dscusson of ts regularty and complement, dscussng the regularty of the matr n great detal. In addton, the nverse of the antsymmetrc trdagonal matr s determned analytcally. b a d a b a 0 0 d 0 0 b, = 0 a b 0 d M = = a ( m ),,,,,, 0 0 a a = = = = 0 0 a b a 0 0 d 0, > a b d : = () How to cte ths paper: Gueye, S.B. (04) Sem-nalytcal Soluton of the D Helmholtz Equaton, Obtaned from Inverson of Symmetrc Trdagonal Matr. Journal of Electromagnetc nalyss and pplcatons, 6,

2 b where,, a 0, and d =. In case where a s zero, the nverson of the matr (M) presents no a dffculty. So we wll focus on the matr () and we wll determne the eact form of ts nverse, (B). We proceed as follows: frst, we determne the determnant of () and gve a very detaled dscusson of ts nvertblty. Therefore, we formulate ts nverse analytcally and eactly. Then, we solve the Helmholtz equaton, wth the fnte dfference method, usng the obtaned nverse matr. ddtonally, we treat the skew-symmetrc trdagonal matr and gve the formula of ts nverse.. Determnant of the Matr ().. Characterstc Equaton and Dscrmnant The calculaton of the determnant of () and the dscusson of the estence of ts nverse consttute a very mportant part of ths work. The determnant of () depends on and s denoted. We defne = and 0 = d. We also defne as follows: d = = d. () d By developng the determnant wth respect to the frst row, one fnds that t follows a second-order lnear homogeneous recurrence relaton wth constant coeffcents [] [3]: = d,. (3) The term s the determnant of the submatr of (), obtaned by elmnatng ts frst row and ts frst column. denotes the determnant of the submatr of order, obtaned by deletng the frst two rows and frst two columns of (). The characterstc equaton of the recurrence relaton, gven by the Equaton (3), s: r d r+ = 0. (4) The resoluton of ths Equaton (4) yelds the epresson of n terms of. The solutons of the characterstc equaton are determned by the sgn of the dscrmnant.. Case =0 : d = ± = d 4. The dscrmnant s zero for two values of d : d = or d =. For ths case, the characterstc equaton d admts one double real soluton: r = r = r =. Then, the general epresson of the determnant s: = r ( + B ), (5) where ' and B' are two constants whch are determned takng nto account the frst two terms of the sequence. The constant ' s obtaned by consderng: = r + B = (6) The constant B' s determned n the followng manner: d = r ( + B ) = d = ( + B ) B =. (7) So the determnant of the matr (), n the case where d = ±, s gven by the followng formula: ( ) +, d = = ( ) ( +, ) d = (8) 46

3 The Equaton (8) s the eact formula of, when the dscrmnant of the characterstc equaton s zero. One remarks that, n ths case, the matr () s regular: ts nverse ests. Ths regularty of () can be deduced from the epresson of. Because s postve and therefore, the determnant cannot vansh..3. Case >0 : d > It corresponds to the case where d b = a >. Then, the characterstc Equaton (4) has two dstnct real solutons: The general epresson of the determnant s for ths case: d ± r ± =. (9) = + (0) r B r + The constants ' and B' can be determned, consderng the values of One gets: = + B = 0 = r B r d + + = It holds: Thus, the determnant of () s obtaned: r+ r = and B = for the frst two orders: 0 and. () () Ths equaton s equvalent to: = r r (3) + + d + d 4 d d 4 = d 4 The determnant of () s dfferent from zero, for the consdered case ( d > ). Then, the trdagonal symmetrc matr () s regular and ts nverse ests. The remarkable Equaton (4) gves the determnant of the matr () n the case where the dscrmnant of the characterstc equaton,, s strctly postve. One can verfy, for eample, that for d = 4 and = 3, we get = 56. Ths value can be obtaned wth Equaton (4) 3 or drectly. n observaton of the determnant, for ths case, allows another formulaton of the formula n Equaton (4). Because, one remarks that the determnant s a polynomal and can be developed. It holds: 3 4 = ; = d; = d ; = d d; = d 3d + ; = d 4d + 3 d; = d 5d + 6d ; = d 6d + 0d 4 d; = d 7d + 5d 0d + ; etc. 8 From the analyss of these polynomal epressons, one can demonstrate usng mathematcal nducton that the determnant gven by the Equaton (4) can be formulated as follows: [ ] = ( ) d, = 0 (4) (5) where [/] est equvalent to ( dv ). Ths latter formula n Equaton (5) s gven n []. But, we prefer the 47

4 formulaton of Equaton (4) for two reasons. The frst s that we look for a matr nverse. Then, t s mportant to know about the annulaton of the determnant. Choosng the Equaton (5) means that we have to search the zero of the polynomals, to know the nvertblty of the matr. Whle the Equaton (4) shows clearly that the determnant does not vansh and thus, the matr () s regular. The second reason to prefer Equaton (4) to Equaton (5) s that programmng the Equaton (4) s more confortable than programmng Equaton (5). Because the latter needs loops for the sum, and t also needs recursons for the bnomal coeffcents..4. Case <0 : d < b It corresponds to the case where < d = <. Then, the characterstc Equaton (4) admts two comple a conugate solutons: d ± r ± =. (6) These solutons r + and r belong to the comple unt crcle. Because ther magntude s equal to unty: r ± =. Therefore, they can be wrtten n the followng manner: wth r = ± θ ± e, (7) atan, 0 d d < < θ = π + atan, d 0 d < < π, d 0 ± ± = (8) Then, the general epresson of s: ( θ) sn ( θ) = cos + B (9) The constants ' and B' are determned consderng and 0. = = et = cos( θ) + B sn ( θ) = d B = d = cot ( θ). 0 One obtans the followng relaton that gves the determnant of (), n the case where d < : Regularty of the Matr () for ( d < ) ( θ) ( θ) ( θ) = cos + cot sn In ths case, the regularty of () has to be studed. One solves: ( θ) ( θ) ( θ) (0) = cos + cot sn = 0 () Ths Equaton () admts solutons θ = θk that nullfy the determnant of the matr (). These solutons are: 48

5 k 4 d k θk = π = atan, k =,,,. () + d k For these values of θ = θk, correspondng to d = dk, the determnant of the matr () s zero and therefore ts nverse does not est. Ths result s very nterestng. Because t shows that the egenvalues λ k of the matr can be epressed n the followng manner: In the treated case ( 0 d ) k λk = d cos π, k =,,,. (3) + < <, the nverse of the matr () does not est for k d = dk = cos π, k =,,,. So any formulaton of the nverse for the consdered case should e- + clude sub-cases where d takes one of the values d k. Because for such values of d, the matr () s not regular. Specal Case d = 0 Ths corresponds to =. In ths case, we have: r = ± ± ; so π θ = ±. The determnant of the matr () s: p π, = p = cos, p,, 3, = = (4) 0, = p+ The Equaton (4) s a very nterestng result. Frst, t gves the eact formula of determnant for the case where d s zero. In addton, t shows that the matr for ths case s regular for even, and non-reversble for odd. So, we can always guarantee the estence of the nverse matr (), takng an even number of mesh nodes. What deserves to be emphaszed s that for odd, 0 s an egenvalue of the matr (). π d = 0 = d = λ and θ = = θ. (5) Ths closes the dscusson of the determnant of a symmetrc trdagonal matr, smlar to the (M). ll cases were studed n a very detaled manner. In each of these cases, the eact value of the determnant of () s gven and ts regularty has been wdely dscussed. 3. Inverse of the Matr Before startng the determnaton of the nverse of the matr (), t s approprate to dscuss ts propertes. The symmetres of () wll be found n ts nverse (B). Frst, the matr () s symmetrc: a = a. So ts nverse s symmetrc: b = b. In addton, () s persymmetrc.e. t s symmetrcal n relaton to ts ant-dagonal. Ths property also appears n ts nverse, (B): b = b +, +. These two propertes show that the matr (B) s determned when one fourth of ts elements s known. Determnng (B) means to determne the cofactor matr of (). Whle these cofactors are obtaned usng determnants of submatrces of (), t s not dffcult to determne them. Because, a detaled work has been done n the prevous secton, concernng the determnant of () and ts submatrces. Thus, t s easy to see that: cof =. In the same way, t holds: cof =. One has cof = ; and for every element of the frst lne of (B), we have: cof = and: + b = b=, =,,,. (6) So the frst and last lnes, and also the frst and last columns of the matr (B) are known eactly usng the 49

6 symmetry and the persymmetry matr (). We remember the prevous secton gves the formulas of all the determnants. The matr (B) beng the nverse of (), ts components satsfy the followng relatons: d b + b = δ b + d b + b + = δ, <, <, b + d b = δ where δ s the Kronecker s delta. From the frst lne of Equaton (7), t s possble to obtan the second column of the matr (B) from the frst column, whch s already known. Then, wth the symmetry of the matr (B), we have: (7) b = d b = b,. (8) Wth the symmetry and the persymmetry of (B), we also have: b,. = b = b, + = b +, (9) The second lne of Equaton (7) allows to fnd the elements of the thrd row and the thrd column of the matr (B): Consderng the Equaton (8), the followng relaton s obtaned: b + d b + b3 = 0, 3. (30) b = d b = b, 3. (3) 3 The second lne of Equaton (7) also allows to fnd the elements of the fourth row and the fourth column of (B): b + d b3 + b4 = 0, 4. (3) Consderng the Equatons (8) and (3), the followng relaton s obtaned: 3 b = d d b = b, 4. (33) 4 3 The analyss of each element of the matr (B) leads to the complete and eact formulaton of ths remarkable matr: b = b,. (34) The Equaton (34), combned wth the Equaton (6), gves: + b = ( ),. (35) The Equaton (35) determnes all the elements of the upper trangle of the matr (B). So the symmetry of the matr allows to get the closed form of (B), nverse of the symmetrc trdagonal the matr (). Thus, (B) s known and each of ts components s gven by the followng equaton: + ( ), b =,, + ( ), > Ths beautful relatons s very mportant. It s an nterestng result to solve any dfferental equaton whose dscretzaton leads to algebrac equatons of the form + + d + = f, =,,,. It s clear that effectve solutons for such equatons est. But, the eact formulaton of ths matr (B) allow us to avod 430

7 nverson methods that use the RHS. However, t deserves to be precsed that the formula of the Equaton (36) s not new. Indeed, t has already been determned n []. But the present study follows another approach and addtonally provdes a deeper and complete dscusson of the regularty of (): ths work completes the study n []. s applcaton, we wll solve the Helmholtz equaton, whch s a very mportant equaton n physcs, usng the matr (B). We could also take the equaton of heat dffuson and the Posson equaton. But we prefer the former, whch corresponds to the wave equaton for harmonc ectaton. 4. pplcaton wth the Resoluton of Helmholtz Equaton Knowng the matr (B) allows to solve all the boundary problems posed n followng manner: c f, ] a, b[ ( a), ( b) + = = a = b s a scalar feld that obeys to the Helmholtz equaton (or to the harmonc heat dffuson equaton). Ths latter corresponds to the wave equaton for harmonc ectaton. The boundary condtons are of frst knd: Drchlet-Drchlet.e. ( a) = a and ( b) = b. The RHS, f, s a specfed functon. The constant k s also known. We consder an one-dmensonal mesh wth + dscrete ponts ( ). Each pont ( ) s defned by b a =, where = = h beng the step sze. We defne, f = f, = 0,,, +. + The applcaton of the fnte dfference method to the Equaton (36), wth the centered dfference appro- O h, leads to the followng algebrac system of equatons [4]-[6]: maton where d = + c h. Thus, one gets n matr form where the vector F s defned by: Thus, t holds ( B) d + h f,,,, (36) + + = = (37) d h f a d h f 0 d h f d 0 4 h f d =, 5 h f h f d h f b : = : = : = F F = h f F = h f F = h f = (39) a, b, and,,3,,. = F. The soluton at pont = Ths can be epressed n the followng form (38) s gven by a smple matr-vector multplcaton: = b F, =,,,. (40) = b F + b F, =,,,. = = + (4) 43

8 whch gves fnally: ( ),,,,. (4) = F + F = = = + Ths Equaton (4) gves the soluton One can also defne at mesh pont. µ = mn (, ) = + and ν = sup (, ) = + + (43) Then, each element b of the matr (B) s gven by: Thus, each soluton at pont + µ µ b = ( ),, =,,,. (44) ( ) can be wrtten: = F, =,,,. (45) µ µ = The Equatons (4) and (45) are two forms of soluton of the Helmholtz equaton. Each of these two forms can be mplemented smply and elegantly n a source code. 5. umercal Results the Dfferent Cases The dfferent studed cases are consdered to llustrate the effcent of the proposed approach that s based on the eact nverson of the mportant matr (). Logcally, ths method s stable robust and very accurate. Because the method of nverson does not use the RHS of the dfferental equaton. For applcatons the value a = 0 and b = have been chosen. The relatve error ε ( ) at each pont s defned by the followng relaton: ε = FDM eact ( ) eact The average relatve error ε ( ), s computed, accordng the formula: 5.. Results for =0 : d = ε = ε (46) = (47) Sub-Case = 0 and d = The sub-case where d = s consdered. Ths corresponds to c = 0 and the dfferental equaton becomes the Posson s one; whch we dscussed n [] [4]. Here, we chose = 0. The results are presented n the Table. 7 It holds, for the consdered case: ε ( 0) Sub-Case = 0 and d = + 4 For ths sub-case, we have d = +. The constante c s dfferent from zero and s: c =. The dfferental h equaton can be an Helmholtz or an Heat Dffuson s equaton or any other dfferental equaton, dscrbed by Equaton (37). The results are presented n the Table, wth = 0. 0 Ths results are very accurate and the average relatve error s: ε ( 0) One remarks that t more accurate than the prevous case, whch dealt wth an ellptc equaton: the Posson s one. 5.. Results for >0 : d > 43

9 5.5 The Equaton (4) gves the formula of the determnant. Takng c =, one gets: d = 7.5. Then, the h results, gven by the Table 3, are obtaned for = 0 : 0 The average relatve error s: ε ( 0) Table. Results for sub-case = 0 and d =. FDM eact ε ( 00 ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 0008 Table. Results for sub-case = 0 and d = +. FDM eact ε ( 00 ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E

10 E E E E E E E E E E E E 00 Contnued E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 000 Table 3. Results for > 0 : d >. FDM eact ε ( 00 ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

11 E E E E E E E E E E E E E E E E Results for <0 : d < Results for = 4 : d = 0 s dscussed prevously, an even s adequate for ths case. We have chosen = 00. Of course, and we have c = n order to get d = 0. The results are shown n the Table 4. h 0 The average relatve error s: ε Results for < 0 and 0< d < 3.5 We have chosen = 00, c =. Thus, we get d =.5. The obtaned results are shown n the Table 5 h below. 0 The average relatve error s: ε ( 00) Results for <0 and < d < 0.7 Here, we chose = 00, c =. Thus, we get d = 0.3. The obtaned results are shown n the Table 6. h 0 The average relatve error s: ε Table 4. Results for < 0 : d <. FDM eact ε ( 00 ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

12 E E E E E E E E E E E E E E E E E E E E 000 Table 5. Results for < 0 and 0< d <. FDM eact ε ( 00 ) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 000 Table 6. Results for < 0 and < d < 0. FDM eact ε ( 00 ) E E E E E E E E E E E E E E E E E E E E E E E E

13 E E E E E E E E E E E E E E E E 000 Contnued E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Inverse of the Trdagonal ntsymmetrc (Skew-Symmetrc) Matr We gve here, addtonally, the nverse of the trdagonal antsymmetrc matr: M b a d a b a 0 0 d a b 0 d = = a, a a b a 0 0 d a b d rgung as we dd wth the matr (), we get the characterstc equaton to obtan the determnant of ( ) : The dscrmnant r : = (48) d r = 0. (49) = d + 4 and s strctly postve. Thus, the determnant of as the Equaton (4), wth the correspondng dscrmnant of the characterstc equaton for ( ) : ( = d + 4) : = d, and for, t holds: 0 =, + + d + d + 4 d d + 4 = d + 4 has the same form One can remark that ( ) s regular for any value of d dfferent from zero. f d s zero and s odd, then the nverse of ( ) does not est. 437

14 Then, the nverse of ( ) that we denote, > B s gven by the followng relaton:, b =,, The correspondng applcatons for the nverse matr ( B ) are dfferental equatons whose dscretzaton leads to algebrac equatons of the form + d = f, =,,,. 7. Concluson + Ths study has gven the sem-analytcal soluton of each equaton dfferental whose dscretzaton leads to algebrac equatons of the form + + d ± = f, =,,,. The estence of the nverse of the dscretzaton matrces s wdely dscussed. The presented approach s stable and gves very accurate results. The consdered boundary problems are Drchlet type. References [] Hu, G.Y. and O Connell, R.F. (996) nalytcal Inverson of Symmetrc Trdagonal Matrces. Journal of Physcs, 9, [] Rosen, K.H. (00) Handbook of Dscrete and Combnatoral Mathematcs. nd Edton, Chapman & Hall/CRC, UK, 79. [3] Epp, S.S. (0) Dscrete Mathematcs wth pplcatons. 4th Edton, Brooks/Cole Cengage Learnng, Boston, [4] Gueye, S.B. (04) The Eact Formulaton of the Inverse of the Trdagonal Matr for Solvng the D Posson Equaton wth the Fnte Dfference Method. Journal of Electromagnetc nalyss and pplcaton, 6, [5] Engeln-Muellges, G. and Reutter, F. (99) Formelsammlung zur umerschen Mathematk mt QuckBasc-Programmen. Drtte uflage, BI-Wssenchaftsverlag, [6] LeVeque, R.J. (007) Fnte Dfference Method for Ordnary and Partal Dfferental Equatons, Steady State and Tme Dependent Problems. SIM, Phladelpha