American Journal of Computational Mathematics, 2013, 3, 5661 doi:10.4236/ajcm.2013.33B010 Published Online September 2013 (http://www.scirp.org/journal/ajcm) Differential Games of Pursung in the Systems with Distributed Parameters and Geometrical Restrictions M. Sh. Mamatov, E. B. Tashmanov, H. N. Alimov Department “Geometry”, National University of Uzbekistan Named After M. Ulugbek, Tashkent, Uzbekistan Email: mamatovmsh @ mail.ru Received 2013 ABSTRACT A problem of pursuit in the controlled systems of elliptic type without mixed derivativ es with variable coefficients was considered. The model of the considered system is described by partial differential equations. The players (opponents) control parameters occur on the righthand side of the equation and are subjected to various constraints. The first player’s goal is to bring the system from one state into another desired state; the second player’s goal is to prevent this from happening. We represent new sufficient conditions for bringing the system from one state into another. The fi nitedifference method is used to solve this problem. Keywords: Pursuit; Pursuer; Evader; Terminal Set; Pursuit Con trol; Evasion control 1. Introduction Some problem formulations in the theory of differential games may be illustrated by motion of two controlled objects, pursuer and evader. Let in the course of motion the objects continuously observe each other and at each time instant correct their motions depending on the in formation about the adversary. Depending on the pur suer’s aim, the problem of pursuit is then formulated as follows: using the information about the evader, at each time instant t select a cont rol such that coincid enc e of the objects’ spatial coordinates is reached as soon as possi ble. The majority of studies consider the case where be havior of the lumpedparameter model described by a system of ordinary differential equations. This scheme encompasses many problems of differential games aris ing in diverse filed of the natural sciences. The mathe matical issues of the differential games describing the lumpedparameter systems were developed in detail. In many applications, however, the lumpedparameter models describe phenomena inadequately. It often turns out that a system which is optimal in the sense of a sim plified model does not use the additional designedin potentialities of control. The distributedparameter mod els obeying the differential equ ations with partial deriva tives offer a better, more adequate description. Use of these equations also gives rise to various game problems of which one is the subject matter of the present paper. It focuses only on the problem of pursuit. Therefore, we make an assumption about the nature of information for this problem. 2. Formulation of the Problem The operated distributed system described by the elliptic equations (see, for example, [1,2]) is considered 22 22 (,) /(,) /((,),(,))axy zxbxy zy fuxyxy , (1) /(,)(,zxyzx)y , (,)xy where (,)zzxy – unknown function, , – continuous functions in with border (,)axy ):0 1, x(,)bxy 0 1} y {( ,xy , (,) y – smooth function on , – external normal. It is supposed that there is a positive constant such that for any the inequal ity, (, )xy (, )ybx ,(, )uuxy , (,) y 2()L – operating func tions is executed from a class . The first (pursuing) player, (pursued or escaping) the player, uP, Q , P and Q – nonempty compacts in disposes of function 1 R (, ) y second of function . The ter minal set (, )uxy 1 1 R s allocated. Definition 1. In a task (1) it is possible – comple tion of (0) prosecutions from “boundary” situation (,) , if exist function (,,)uxyP , Q , (, )xy , such that for any function 0(,) yQ , (, )xy 0 ((,),xy the solution of a task (1) where 0 zx(,)t ,)xyu u , 0(,) y , gets on a set 1 M , at some (, ) y , 01 (, ):(, ) yzxy IM where (1I,1) . Decompose the Euclidean space of variables 2 R (, ) y by the planes i ih , , and 1/hr0,1, ,i j yjl , 1/l , 0,1, 2,,j into parallelepipeds Copyright © 2013 SciRes. AJCM
M. Sh. MAMATOV ET AL. 57 (, ){(,) :(1),(1) } ij i yih xihjlyjl , and r being some natural numbers. The points (, ) ij y hl belonging to a set are the nodes of the grid . Each node has its neighbors. If all these neighbor nodes also belong to the grid, then the node hl (, ) ij y is referred to as “internal”, otherwise, (, )ihjl(, ) ij y 2 1 (),h 22 (), /, ,,, , , , ij is called the “boundary” node. The set of all boundary nodes is called as border of net area and is designated through . h 2 11 2 11 22 11 22 1 /(,)((,)(,))/2 (), /(,)((, )(, ))/2(), / (,)((,)2(,)(,))/ / (,)((,)2(,)(,))/ iji ji j ij ijij iji jijij ijijij ij zxxyzx yzx yhOh z yxyzxyzxylOl z xxyzxyzxyzxy zyx yzx yzx yzx yl ,ij z ,1 ,,1,,,1 ,,1 (2)/ (2 0,1,,1; 0,1,,1. ijijij ijijijijij azz zhbzz z irj (, ): i jij Replace the internal nodes of the derivatives (1) dif ferential secondorder accuracy of approximation ratios with formulas 2 O Ol ij f , ij ij h 22 )l , Substituting these ratios in (1), having rejected an error of approximation of derivatives, we will receive the dif ferential equations for unknown (2) where the following designations of values of coeffi cients and the right part in a hub ya ,, , ij ij b dc f ,((,),(,)), (,) ijijijijh ffuxy xyxy , for example are entered l ,ij z ,ij z 1,0,0, 0,1,0, ,1,,1,, , ,1,0,0 ,1,0,0 ,,1 ,,,1, ()/()/2, (0,1, ()/()/2, ()/()/2, (0,1,2, ()/()/2 jj jjjj rjr jrjrjrjrj ii iiii ii iiii zz hzzj zz hz z zzl zzi zz lzz 2(r 21) 1)r Ratios (2) contains except unknown in internal nodes also unknown on border of net area. For boundary nodes we will write down a ratio ,, , ) (3) Thus, we will receive system of r ,1i z ,ij z ,i z equa tions with the same number of unknown . Using boundary conditions (3), we will express , through , ,0i z,1i z ,1,0,0 ,0,0,0 ,,,,1, (2)/(2)2 (2 (2)/(2)2 (2 ii iii iiiii zl lzll zl lzl . Let's have , ). i i ) l z z (4) Using these ratios, we will exclude in system (3) un known ,1i, ,i . If to enter designation 2 /hl 2 1,00 ,0,11,0,0 1, ,1, ,11,, 1, 1,2,, 11, 1, (2 2), 2(1)(1,2,, 2), (2 2) (0,1,2,,1), iiiii ij ijij ijijij ii iii zkzzz zz zzzF j zz kzz ir , we will receive system 1 i i F F (5) where 22 ,0 0,0,0, 2 ,1 ,1,, 2/(2); (1,2,, 2); 2(2). ii iiiji ii ii , j hfllF hf j Fhf l l (6) This system can shortly be written down in a look 11 , (0,1,2,,1). iiii i zAzzFi r (7) where ,0,1, 1,0,1, 1 (,,,); (,,,) iiiii iii zzzzFFF F ， ,0 , 2(1 )000 2(1 )00 02(1)00 000 2(1) i i i k A k (8) Boundary conditions (3) and (5) can be copied in a look 1,0,0,0,0,0, 0, 0,0, 1, , ,,, , ,, , (2)/2(2 )/(2) (0,1,2,,1) (2 )/(2 )(2)/(2 ) (0,1,2,,1) jjjjjj jj j r jrjrjrjrjrj rjrjrj zh hzhh kz yj zh hzhh kz yj ， ， (9) where 0, 0, 0,0,0, 0, ,,,,, (2)/(2) (2)/(2) (2)/(2) (2)/(2). jjjjjj rjrjrj rjrjrj khhyh h khhyh h , ；； ； (10) Having put 00,00,1 0,1,0,1,1 0,0 0,1 0 0, 1 ,0 ,1 1 ,1 (,,,); (,,,). 000 0 0000 ; 000 0 000 0 0000 , 000 0 rrr r r r r yyyyyyy y k k X k k k X k (11) it is possible to write down systems (9) in such look: 1000 11 ,. rr zXzy zXzy r (12) Finally we have the following system of the equations: 1000 11 11 (1,2,,1) iiiii rrr zXzy zAzzF ir zXzy ， 。 (13) Instead of game (13) we will consider more the gen eral game described by system of the equations 0001 0 11 1 (,) 11 nnnnnnn nn NNNN N Cz Bzf AzCzBzfunN AzCz f ， ，， ， (14) where m n zR , 0,nN, ,, nnn CB – constant square matrixes, mm , nn u – operating parameters, – n u Copyright © 2013 SciRes. AJCM
M. Sh. MAMATOV ET AL. 58 prosecution parameter, n – beanie parameter, nn uP R, q nn QR , n and – nonempty sets; n Pn Q – the set function displaying q RR in . Besides, in the terminal set is m R m R allocated. Definition 2. We shall say that from “boundary” situa tion 0 (, ) N f it is possible to complete pursuit for steps if from any sequence N 12 1N ,,, of the values of evasion controls it is possible to construct a sequence 121 of values of the pursuit control values such that the solution ,, , N u uu 012 ,z1 {,,,, } NN zzz z of the equation 00 11 1 nn NN B z 010 (,) 11 nnnnn nn NN N Czzf AzCzBf u AzCz f ， ，， . n N (15) Gets on :i zM for some . Thus for finding of value i n u it is allowed to use values n and n z. Note that the type of systems (14) is difference schemes for elliptic equations of second order with variable coef ficients in any field of any number of dim ensions [314]. Solution of problem (14) will be sought in the form 11 , 1, 0, nnn zz nN 1n2, ,N (16) where 1n mm – uncertain while a square matrix of the sizes , and 1n – a vector of dimension m. From a formula (16) and the equations of system (14) for there are recurrent ratios for calculation of matrixes 11nN n and vectors n . Really from a formula nn z (16) 1n z n substituting it in (14) we will receive ]. nn 1 1 11 () ,), 11; ()(, ; ()(),) nnn nn nnnn nnnn nnnn nnnnn nnnn AzzBz f nN CA zzfu zCABz CAfA 1 n n n n C B ( ) [( n n nn u A u 1 () () 1, 2, nn n 2,, ) nn N A Equating now the right parts of the last and (16) equali ties we will receive 1 1 1 , 1, [(, , . nn n nnnnnn CAB n CA fu nN 1; ], Further from (16) and the equations (14) for 0,nN , there are the initial values 1 , 1 and , allowing beginning the account on recurrent ratios. From (14) and (16) for we will have N z 11 , 0n 11 1 00 zC 00 00 , zCBfzz1 10 CB 0 And, therefore 11 1 00 , .Cf In the same way for we have n N N () NNNNN N zCzf N A N or 1 ()( NN NNN zC fA). Uniting, we will write out final formulas 1 1 11 0 (), 1,2,,, nnnnn CAB nNCB 0 (17) 1 1 1 10 0 ()((,) 1,2,,1. , nnnnnnnnn CA fuA nNCf ), (18) 11 1 1 (, ), 1,2,,0, nnn nnn NN zzu nN Nz (19) It is clear, that if in game (17), (18), (19) n zM that in game (14) too game comes to the end. Therefore fur ther instead of game (14) we will consider discrete game described by system of th e equations (17), (18), (19). Before giving determination of stability of algorithm (17), (18), (19), we will provide some data from linear algebra. Let A – any square matrix and  mmm be norm of a vector in , then the norm A is defined by equal ity m R 0 sup/ . mm x Ax x For a case of Euclidean norms in we have m R  A , where – maximum on the module own value of a matrix A . Without the proof we will give the following known lemma (see [15]). Lemma 1. Let for some matrix norm the square matrix meet a condition  1 q . Then there is a matrix 1 ()EA and  1 ) 1/(1EA ( )q . Let's say that the algorithm is steady if the assessment  1 j for 1jN is carried out. Lemma 2. If C for 0jN – no degenerate ma trixes and and B – nonzero matrixes for 1j 1N also are satisfied conditions 11 0 0 11 1, 1, 1, 11. NN jj jj CB CA CA CBjN And at least in one of inequalities the strict inequality takes place, there are return to the . jj CA matrix and  1 j , here 1 10 0 СB , 1 1(), 1 jjjjj CA BjN1. Proof. 1 10 0   1CB , suppose, that  1 j also we will show 1  1 j . After a course the proof of this fact we will receive existence of a matrix 1 () jjj CA . Really from conditions of a lemma we will have 11 11 1 1. jjj jjj jjjj CACACA CB As 1 jj CA square matrix that owing to a lemma 1 there are return to 1 jj ECA and jj CA matrixes and 1 ) 1 1/ ( jjj j CBECA . From here and from (17) we will receive 111 1 111  () () 1. jjjjj jjjjj ECA CB ECA CB j The proof of the lemma is complete. Copyright © 2013 SciRes. AJCM
M. Sh. MAMATOV ET AL. 59 3. Main Results Everywhere further it is supposed that 01 MM , where 0 – linear subspace , 1 m R – a subset a subspace, – orthogonal complement of 0 L in . Denote we will designate a matrix of orthogonal design from on . n R m RL Let , (0){0}W 1 1 0 11 ()( ,) ()(), 1. NkiNki k kNki NkiNkiNki iQ Wk P WkWkkN (20) Theorem 1. Let N be the smallest of the numbers k, such that 1(). NkNkN N zWk 1 (21) Then from “boundar y” situation 0 (, ) N f it is possible to complete pursuit for N steps. Let now , 21 (0)WM 11 11 22 11 22 111 (1)[(0)(,)], ()[ (1)(,)] Nk Nk NN Nk NkNk Q Nk NkNNNN Q WWP Wk WkP 1 (22) Theorem 2. If N be smallest of those numbers, for each of which takes place inclusion k () 1 zW k NkNkNN 2 (23) that of “boundary” situation 0 (, ) N f it is possible to complete pursuit for N steps. Let 1 01 10 (),,,:0,1 k kki i i ], and 1 11 0 313 () (()) (,) 0, (0), ()(()), 0. Nki Mki k k k iNkNki NkiNkiNki iQ k W MP kN WMWkW kN (24) Theorem 3. If 1 – a convex set and N be small est of those numbers . For each of which inclusion takes place k 1(). NkNkN N zWk 3 (25) That of “boundary” situation 0 (, ) N f it is possible to complete pursuit for N steps. It is easy to be convinced [15] that the solution of differential task (2) meets to the solu tion ,ij z of an initial task (1), the following assessment of speed of conver gence takes place 2 ,12 ( ), hl hli j zz KhKl 2 (26) where – values of the exact decision a task (1) in grid functions, () hl z hl – spaces of net functions,  hl – is its norm and, 1 and 2 constants. Theorem 4. Let in an inequality (26) 22 12 Kh Kl , and in game (13) from a “boundary” situation 0 (, ) N f 0 (, ) N yy /(zx completion of prosecution that is definitions 2 be possible. Then in game (1) fro m “boundary” situation ,)(,yzx)y , it is possible to complete pursuit that are definitions 1. (, )xy 4. Proof of Theorem Proof of Theorem 1. Let 12 1 ,,, N , i Qi , 1iN1 – any sequence. Instead of inclusion (21) we will consider other inclusion equivalent to it 11 11 111 (1) (, NN NkNkN N Nk NkNNNN Q zWk P 1 ) Means, exists 1N a 11 111 (, ) NN NNkNkNNN Q aP 11 . N Such that 11 (1) . NkNkN NN zWk a 1 (27) Now control of the pursuing player 1N u, the relevant control of the escaping player 1N , we will construct as the solution of the following control 1111 (, ) Nk NkNNNNN ua 1 . It is clear, that the equation has the decision. From here owing to (27) we have 1 111 (1)(,) NkNkN N Nk NkNNNN z Wk u 11 We write down this inclusion in other look. 11 111 [(,)](1) NkNkNNNNNN zu Wk. (28) As a result from equalities (18) and (28) we will re ceive 1111 (1) Nk NkNN zWk (29) Done above a reasoning allow us to construct on the set control 1N providing inclusion (29). If now the control 2N becomes known that, we above can receive in the stated way control 1N u providing inclusion 1221 (2). Nk NkNN zWk Repeating this process, further we can construct step by step control i u, proceeding from becoming known controls i , therefore, that in any step inclusion takes place 111 (0). Nk zW It means that Copyright © 2013 SciRes. AJCM
M. Sh. MAMATOV ET AL. 60 1Nk z As we set out to prove. Proof of Theorem 2. Let 121 ,,, N , ii Q , – any sequence. For concrete 1iN 11N owing to (22) and (23) we will receive inclusion 12 111 (1) (, ) NkNkN N Nk NkNNNN zWk P 1 (30) Now as 1N u we take that element from 1N P for which inclusion (30) remained. Then we will receive 12 111 (1) (, ) NkNkN N Nk NkNNNN zWk u 1 From this it follows that 11 112 [(,)]( NkNkNN NNNN zu Wk1). And therefore, owing to (19) we have 1112 (1). Nk NkNN zWk If now the control 1N becomes the stated way known that we above us can constru ct control 1N u pro viding inclusion 1222 (2). Nk NkNN zWk Further arguing similarly in any step we will receive 12 1 (0) , Nk zW that is 1. Nk z The theorem is proved completely. Proof of Theorem 3. Instead of inclusion (25) mean ing (24) we will consider inclusion equiv a lent to it 1(()). NkNkN N zW Existence 1 01 10 (),,,,0,1 k kii i follows from (24). From here follows 11 2 10 11 1111 1 [(, [(,) Nki Nki NN k NkNkN N iQ iNkNki NkiNkiNki kNkNNNN Q z P P )] ]. (31) Let now 12 1 ,,, N , ii Q , – any se quence. Owing to (31) exists such that 1iN 1N a 1 11 111 11 2 10 11 [ [( NN Nki Nki NkNkNNN Q k NkNkN N iQ iNkNki NkiNkiNkiN P z Pa 1 1 (,)], ,)] N (32) Therefore, controls 1N u we will construct as the so lution of the following equation 111111 1,11 (,), . kNkNNNNNk muam Further owi n g t o (32) we have 2 10 11 11111,1 [( (, ). Nki Nki k NkNkN N iQ iNkNki NkiNkiNki NkNN NNkk z P um ,)] It is equivalent to the fo llowing 11 1111,1 2 11 0 [(,)] [( Nki Nki NkNkNN NNNNkk k iNkNki NkiNkiNki Q i zu m P ,) Therefore owing to (32) we have 11111,1 2 11 0[( Nki Nki Nk NkNNkk k iNkNki NkiNkiNki Q i zm P ,)]. In the same way, if the control 2N becomes the stated way known that we above us can construct con trols 2N u providing inclusion 3 1 221122 0 11 [(, NkiNki k Nk NkNNkkkkQ i iNkNki NkiNkiNki zmm P )] etc. Thus, we will receive 3 1 221122 0 11 [(, NkiNki k Nk NkNNkkkk Q i iNkNki NkiNkiNki zmm P )] from here we receive 1. Nk z The theorem is proved completely. Proof of Theorem 4. Let in game (13) one be able to complete the pursuit from “boundary” situation 0 (, ) N ff 0 (, ) N yy in steps. Then, it follows from Defini tion 2 that from any sequence N 01 , ,...,1 , , Nk Q 0kN1, of the evasion control it is possible to con struct a sequence 01 1 , ,...,,, Nk uuuu P of pursuit control such that the solution 01kN, 01 1 ( ,,...,,) NN zzz z of the equation 100 zXzy 0 , , 11nnn zzF 1n nn zA 1N , 11 z N zX NN y , for some hits dN:d z . Let now in game (2) (,),(, )xy Q xy 2() , be an arbitrary control of an evader from the class L . With the knowledge of the evader control (, ) y , it is possible to determine ,ik as the values of this func tion at the node points of the grid , that is, hl 1, 2,1, ( ,,...,). kkk krk Whence it follows that in virtue of Theorem 4 we can construct the pursuer control in game (13) providing completion of pursuit 1, 2,1, ( ,,...,). kkkkrk uu uuu Now in game (2) we construct the pursuer control (, )uuxy as follows: ,, (, ){:(1), iki ki uxyuuih xih 0,1,...,1,(1) ,0,1,...,1}irklyklk uP . Obviously, and 2 (, )()uxy L. By substituting (,) y and (, )yuux in (2), we obtain a differen Copyright © 2013 SciRes. AJCM
M. Sh. MAMATOV ET AL. Copyright © 2013 SciRes. AJCM 61 [2] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, “Lineinye I Kvazilineinye Uravneniya Parabolicheskogo Tipa,” (Linear and Quasi linear Func tions of Parabolic Type), Moscow, Nauka, 1967. tial equation. Similarly, by substitu ting ,ik and ,ik u in (3), we obtain a grid equation approximating equation (2). Let () hl z be the value of the exact solution corre sponding to the controls (, ) y and (, )uuxy of problem (2) at the nodes of the grid, , hli k z be the solu tion corresponding to the controls ,ik and ,ik u of the difference problem (3). Then, we obtain from (13) and the condition of Theorem 4 that [3] V. A. Il’in, “Boundary Control of String Oscillations at One End with Other End Fixed, Provided that Finite En ergy Exists,” Dokl. Ross. Akad. Nauk, Vol. 378, No. 6, 2001, pp. 743747. [4] V. A. Il’in and V. V. Tikhomirov, “Wave Equation with Boundary Control at Two Ends and Problem of Complate Oscillation Damping ,” Diff. Uravn., Vol. 35, No. 5, 1999, pp. 692704. 2 ,12 () . hl hli k zz KlKh From this fact and ,ik zM1 , we obtain , () , hli k zz S , () ,() hli khl zSzzSM [5] Yu. S. Osipov and S. P. Okhezin, “On the Theory of Dif ferential Games in Parabolic Systems,” Dokl. Akad. Nauk SSSR, Vol. 226, No. 6, 1976, pp. 12671270. 1 n , which proves the theorem. 5. Conclusions [6] F. L. Chernous’ko, “Bounded Controls in Distrib utedparameter Systems,” Prikl. Mat. Mekh., Vol. 56, No. 5, 1992, pp. 810826. Thus, to solve the game problem of pursu it in th e fo r m (1) we pass to the discrete game (13) or (14), and Theorems 13 establish the sufficient condition for such problems. Theorem 4 establishes the sufficient conditions for solv ing the problem of pursuit (1). Here, the difference (see Section 3) plays the main part in the solu tion of problem and implies that the solutions of the grid equation (2) are stable. , () hli j zz [7] N. Satimov and M. Sh. Mamatov, “On a Class of Linear Differential and Discrete Games between Groups of Pur suers and Evaders,” Diff. Uravn., Vol. 26, No. 9, 1990, pp. 15411551. [8] N. Satimov and M. Tukhtasinov, “On some Game Prob lems in the Distributed Controlled Systems,” Prikl. Mat. Mekh., Vol. 69, No. 6, 2005, pp. 9971003. The problem of stability of the grid equation (2) lies in determining the conditions under which the numerical error tends to zero with growing j uni formly in all , or at least remains bounded. , () ijhl ij pz z ,0i , i [9] N. Satimov and M. Tukhtasinov, “On some Game Prob lems in Controlled Firstorder Evolutionary Equations,” Diff. Uravn., Vol. 41, No. 8, 2005, pp. 1114 1121. [10] M. Sh. Mamatov, “On the Theory of Differential Pursuit Games in Distributed Parameter Systems,” Automatic Control and Computer Sciences, Vol. 43, No. 1, 2009, pp. 18. doi:10.3103/S0146411609010015 Equation (2) is called stable if the round off errors generated in the course of calculations have tendency to decrease or at least not to increase. Otherwise, the accu mulated errors may reach a value such that the numerical solution has nothing in common with the exact solution of the grid problem (2). It goes without saying that such unstable grid equations cannot be used for nu merical solution of the differential games. () hl z [11] M. Sh. Mamatov, “About Application of a Method of Final Differences to the Decision a Prosecution Problem in Systems with the Distributed Parameters,” Automation and Remote Control, Vol. 70, No. 8, 2009, pp. 13761384. doi:10.1134/S0005117909080104 [12] M. Tukhtasinov and M. Sh. Mamatov, “On Pursuit Prob lems in Controlled Distributed Systems,” Mathematical notes, Vol. 84, No. 2, 2008, pp. 273280. Theorems 14 are easily generalized to a wider class of differential games, for example, when 2 1212 12 2 1(,,...,)(( ,,...,),(,,...,)) n nn z axxxfuxxx xxx x n [13] M. Tukhtasinov and M. Sh. Mamatov, “About Transition Problems in Operated Systems,” Diff. Uravn., Vol. 45, No.3, 2009, pp. 16. with discontinuous coefficients. [14] M. Sh. Mamatov and M. Tukhtasinov, “Pursuit Problem in Distributed Control Systems,” Cybernetics and Systems Analysis, Vol. 45, No. 2, 2009, pp. 297302. doi:10.1007/s105590099100x REFERENCES [1] O. A. Ladyzhenskaya, “Kraevye Zadachi Mate maticheskoi Fiziki,” (Boundary Problems of Mathemati cal Physics), Moscow, Nauka, 1973. [15] G. I. Marchuk, “Metody Vychislitel’noi MateMatiki,” (Methods of computational Mathematics), Moscow, Nauka, 1989.
