The Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method

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1 Journal of Electromagnetc Analyss and Applcatons, 04, 6, 0-08 Publshed Onlne September 04 n ScRes. The Exact Formulaton of the Inverse of the Trdagonal Matrx for Solvng the D Posson Equaton wth the Fnte Dfference Method Sergne Bra Gueye Département de Physque, Faculté des Scences et Technques, Unversté Chekh Anta Dop, Dakar-Fann, Sénégal Emal: sbragy@gmal.com Receved June 04; revsed 5 July 04; accepted 8 August 04 Copyrght 04 by author and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract A new method for solvng the D Posson equaton s presented usng the fnte dfference method. Ths method s based on the formulaton of the nverse of the trdagonal matrx assocated wth the Laplacan. Ths s the frst tme that the nverse of ths remarkable matrx s determned drectly and ly. Thus, solvng D Posson equaton becomes very accurate and extremely fast. Ths method s a very mportant tool for physcs and engneerng where the Posson equaton appears very often n the descrpton of certan phenomena. Keywords D Posson Equaton, Fnte Dfference Method, Trdagonal Matrx Inverson, Thomas Algorthm, Gaussan Elmnaton, Potental Problem. Introducton The fnte dfference method s a very useful tool for dscretzng and solvng numercally a dfferental equaton. It s effectvely a classcal method of approxmaton based on Taylor seres expansons that has help durng the last years theoretcal results to gan n accuracy, stablty and convergence. In fact, ths method s very useful for solvng for example Posson equaton. Ths ellptc equaton appears very often n mathematcs, physcs, chemstry, bology and engneerng. In one dmenson, the resoluton leads to a trdagonal matrx n the case of centered dfference approxmaton. Ths matrx, whch s dagonally dom- How to cte ths paper: Gueye, S.B. (04) The Exact Formulaton of the Inverse of the Trdagonal Matrx for Solvng the D Posson Equaton wth the Fnte Dfference Method. Journal of Electromagnetc Analyss and Applcatons, 6,

2 nant, can be nverted wth methods such as Gauss elmnaton, Thomas Algorthm Method []. These techncs are powerful and very effcent. We proposed here, a new and drect method of nverson of ths trdagonal matrx ndependently of the rghthand sde. For Drchlet-Drchlet boundary problems, ths nnovatve method s faster than the Thomas Algorthm. It gves better accuracy and s far more economcal n terms of memory occupaton. Frst, the fnte dfference method s presented for the D Posson equaton. Secondly, the propertes of the matrx assocated wth the Laplacan and ts nverse are dscussed. Then, the nverse matrx s determned and ts propertes are analyzed. Thus, verfcaton s done consderng an nterestng potental problem, and the sensblty of the method s quantfed.. Fnte Dfference Method and D Posson Equaton We consder a functon ( x) whch satsfes the Posson equaton ( x) = f ( x), n the nterval ],[ ab, where f s a specfed functon. ( x) fulflls the Drchlet-Drchlet boundary condtons ( a) = a and ( b) = b. We consder an one-dmensonal mesh wth + dscrete ponts ( x ). Each pont ( x ) s de- ( b a) fned by x = a+ x, where x = = h beng the step sze. We defne ( x), f = f ( x), + = 0,,, +. We have chosen the centered dfference approxmaton ( O( x )), n ths work, for the fact that t gves a trdagonal, dagonally domnant, and symmetrc matrx. Consderng all the above mentoned crtera, one can rewrte the D Posson equaton n a set of algebrac equatons: + = = () + h f,,,,. One gets a lnear system of equatons, whch can be wrtten n a matrx form [] h f a h f h f h f = h f h f h f b : = A : = : = F Thus, solvng the D Posson equaton means to nvert the negatve defnte, and regular -matrx = B =, s also symmetrc. Both matrces have the followng propertes: A ( a j ). Its nverse, that we noted ( b j ) and where, = j aj =, j = 0, j > b + b = δ j bj bj + bj+ = δ, < j <, b b = δ j δ s the Kronecker s delta.. The Inverse of Matrx A From (4), we derve successvely the followng nterestng relatons: () () (4) 04

3 ( ) b = j b + j b and b = j jb + + j (5) wth (5), one sees that the matrx B s entrely determned f the term b s known. Ths term can be determned by observng the behavor of B for dfferent values: It holds From (5) and (6), we get b b = b + = j + = + ( j ) ( ) ow, the matrx B s completely and ly determned. B = ( b );, j =,,, wth B = b j ( ) j, j + = ; ( j ), j + < ( ) ( ) ( j ) ( ) ( ) ( ) ( j ) ( ) ( ) ( ) ( j ) ( ) ( ) ( ) ( ) j 4 6 j j j ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The soluton of the D Posson equaton s obtaned wth a smple, extremely fast matrx multplcaton: = BF. Thus, the numercal resoluton of the D Posson equaton whch s an nterestng topc n physcs and engneerng s made easy and very accurate. Analyss A frst analyss of the matrx ( ) B let us beleve that, ths new method possesses an algorthm complexty of ( O ) and the one of Thomas s ( O( )) []. B shows that the complexty brought by the Thomas method s largely ( O( )), whch s stuated between the Gauss elmnatons ( ) A deeper analyss of the matrx ( ) mproved n ths study. In addton, one can see a close lnk between ts row vectors and column vectors. The matrx ( B ) s also persymmetrc: b = b + + j j, All the nformaton about t, can be found n the upper trangle (n gray color, see Fgure ). Further, we can even fnd very nterestng relatons n ths matrx whch can help refnng the fnal soluton. That s what we effectvely dd, and one can see a drect soluton for at the pont x, whch can be expressed by = h (6) (7) (8), f (9) = 05

4 Fgure. Matrx symmetres. Also a drect soluton for at the pont x s: = h f + ( ) f = (0) Generally, a very mportant recurrence relaton can be obtaned, whch gves all solutons: whch s equvalent to: k k= h ( k+ ) f + ( k) ( ( )) f, = = k+ k = 0,,, k k = h k+ f + k f k = = = k+ ( ) ( ( )),,,, () Ths very nnovatve Equaton () gves drectly and accurately all the soluton that we are lookng for. It proves that our method s drect, faster than the one of Thomas s n ths context and gves as well better accuracy. Furthermore, t s far more economcal n terms of memory occupaton. Ths s due to the fact that the matrx ( B ) does not necesstate to be generated. A programmer does not need to declare nor to defne the matrx ( B ) n hs code. In concluson to ths, we can say that the matrx ( B ) s the key of ths effcent new method. Ths matrx ( B ), whch s the nverse of matrx ( A ), s determned explctly, drectly, and ndependently of the rght-hand sde of the Posson equaton..b.: One can prove usng mathematcal nducton that det ( A ) = ( ) ( + ). It holds for the (, j ) cofactor of A : ( ) + CofAj = j ( ), j. We call the matrx B Bra s Matrx. 4. Verfcaton wth a Potental Problem We consder a scalar potental ( x), defned n [0, ], whch satsfes ( x) ( x) = = f ( x) = cos ( x ). ( x) fulflls the followng boundary condtons: ( 0) ( ) π x = =. The soluton s ( ) 0 cos π x x x ( x) = π 4 Wth the fnte dfference method, we take = 00, = ( ) = cos π f f x x. The soluton s x = h = +, x x =, ( x ) () (), and 06

5 h f h f h f 4 h f4 = 5 0 h f, h f h f 00 (4) Dscussons We defne the varable ε ( 00). Generally, we have, whch s the relatve error at pont ε FDM ( ) = x for ( 00) We can also defne the average value of the relatve error for a gven : ( ) 5 ( 00) =. FDM represents ε. For = 00, t s: ε We obtan the followng results, presented n Table. The table shows that the soluton s very accurate. otwthstandng that we have been nterested n determn- ε for dfferent values. ng the sensblty of the proposed method. Effectvely, we have plotted ( ) We obtan a hyperbola, whch can be predcted as proportonal to ( ) Ths curve s ftted wth a functon whch can be defned as ( ) α α Trunc = h =, ( + ) where α We obtan two curves represented n Fgure. Table. Results and relatve error. h = + = x. (5) (6) x (00) FDM ε 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

6 Fgure. Sensblty. We realze that the average relatve error ( ) ( 4) h ( c) ( 4). ( c) lowng manner C ) whch belongs to the nterval [ ab, ]. For our gven functon ε behaves lke a truncaton error that we express n the fol- s the fourth order dervatve of the functon n a pont (here and also the results from the fttng, we have the followng relatons []: α h 4π ε ( ) α h = <, + ( ) Ths proves that the method s very accurate, naturally stable, robust, quck and precse. 5. Conclusons Ths paper has provded a new mproved method for solvng the D Posson equaton wth the fnte dfference method. Accurate results have been obtaned wth a sensblty found to be as the functon of ( + ). In fact, the nverse of the trdagonal matrx, whch s assocated wth ths dfferental equaton, s determned drectly, ly, and ndependently to the rght-hand sde. Thus, a new formulaton of the soluton s gven wth an algorthmc complexty of O(). Wth ths nnovatve method, the D Posson equaton, wth Drchlet-Drchlet boundary condton s solved, wth only one programmng loop. Ths new approach provdes also gan n accuracy and economy n memory allocaton. A future work can consder eumann or mxed boundary condtons. Acknowledgements I would lke to thank my colleagues Dr. Chekh Mbow and Dr. Kharouna Talla for beneft dscussons and remarks that contrbute to mprovng the qualty of ths paper. References [] Conte, S.D. and de Boor, C. (98) Elementary umercal Analyss: An Algorthmc Approach. rd Edton, McGraw- Hll, ew York, [] Leveque, R.J.E. (007) Fnte Dfference Method for Ordnary and Partal Dfferental Equatons, Steady State and Tme Dependent Problems. SIAM, [] Mathews, J.H. and Kurts, K.F. (004) umercal Methods Usng Matlab. 4th Edton, Prentce Hall, Upper Saddle Rver, -5, 9-4. (7) 08