DOCSLIB.ORG

Solutions to the Exercises

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Solutions to the Exercises Appendix A Solutions to the Exercises For many exercises extensive calculations are necessary that are difficult to perform by pencil and paper. It is recommended to apply the userfunctions provided on the website www.alltypes.de in those cases; a short description is given in Appendix E. Chapter 1 00 0 1.1. For y C a1y C a2y D 0 the answer is ˇ ˇ ˇ ˇ ˇ ˇ 1 ˇ y y ˇ 1 ˇ y0 y0 ˇ ˇ y y ˇ a D ˇ 1 2 ˇ ;aD ˇ 1 2 ˇ W .2/ D ˇ 1 2 ˇ : 1 .2/ ˇ 00 00 ˇ 2 .2/ ˇ 00 00 ˇ where ˇ 0 0 ˇ W y1 y2 W y1 y2 y1 y2 000 00 0 For y C a1y C a2y C a1y D 0 the answer is ˇ ˇ ˇ ˇ ˇ y y y ˇ ˇ y y y ˇ 1 ˇ 1 2 3 ˇ 1 ˇ 1 2 3 ˇ a D ˇ 0 0 0 ˇ ;aD ˇ 00 00 00 ˇ ; 1 .3/ ˇ y1 y2 y3 ˇ 2 .3/ ˇ y1 y2 y3 ˇ W ˇ 000 000 000 ˇ W ˇ 000 000 000 ˇ y1 y2 y3 y1 y2 y3 ˇ ˇ ˇ ˇ ˇ y0 y0 y0 ˇ ˇ y y y ˇ 1 ˇ 1 2 3 ˇ ˇ 1 2 3 ˇ a D ˇ 00 00 00 ˇ where W .3/ D ˇ 0 0 0 ˇ : 3 .3/ ˇ y1 y2 y3 ˇ ˇ y1 y2 y3 ˇ W ˇ 000 000 000 ˇ ˇ 00 00 00 ˇ y1 y2 y3 y1 y2 y3 1.2. Go to the ALLTYPES user interface and define e1:=Df(z 1,x)-z 2; e2:=Df(z 2,x)-z 3+a 1*z 2+a 2*z 1; e3:=Df(z 3,x)-a 3*z 1+a 1*z 3; T==|LDFMOD(DFRATF(Q,fa 1,a 2,a 3g,fxg,GRLEX), fz 3,z 2,z 1g,fxg,LEX)|; JanetBasis(fe1,e2,e3g|T|); The result of Example 1.1 is returned. The term orders T==|LDFMOD(DFRATF(Q,fa 1,a 2,a 3g,fxg,GRLEX), fz 1,z 2,z 3g,fxg,LEX)|; JanetBasis(fe1,e2,e3g|T|); F. Schwarz, Loewy Decomposition of Linear Differential Equations, Texts & Monographs 181 in Symbolic Computation, DOI 10.1007/978-3-7091-1286-1, © Springer-Verlag/Wien 2012 182 A Solutions to the Exercises T==|LDFMOD(DFRATF(Q,fa 1,a 2,a 3g,fxg,GRLEX), fz 3,z 1,z 2g,fxg,LEX)|; JanetBasis(fe1,e2,e3g|T|); yield the associated equations for z3 and z2 respectively. 1.3. Let the given fourth-order equation be 0000 000 00 0 y C a1y C a2y C a3y C a4y D 0: (A.1) In addition to the functions z1, z2,andz3 defined in Example 1.1 the functions ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 0 0 ˇ ˇ 00 00 ˇ ˇ y1 y2 ˇ ˇ y1 y2 ˇ ˇ y1 y2 ˇ z4 D ˇ 000 000 ˇ ; z5 D ˇ 000 000 ˇ ; z6 D ˇ 000 000 ˇ : y1 y2 y1 y2 y1 y2 are required. They obey the system 0 0 0 0 z1 D z2; z2 D z3 C z4; z3 D z5; z4 D z5 a1z4 a2z2 C a4z1; 0 0 z5 D z6 a1z5 a2z3 C a4z1; z6 Da1z6 C a3a3 C a4z2: A Janet basis in lex term order with z1 ::: z6 has the form X5 X5 .VI/ .k/ 0 .k/ z1 C rk.a1;:::;a4/z1 ; z2 D z1; and zi D fi;k.a1;:::;a4/z1 kD0 kD0 for i D 3;:::;6.Therk and fi;k are differential functions of the coefficients a1;:::;a4. In order to determine the coefficients of a second-order factor (compare the discussion in Example 1.3) it suffices to find a solution with rational logarithmic derivative of the first equation for z1. 1.4. Substituting y D z into the given equation yields Á Á 0 00 0 z00 C 2 C p z0 C C p C q z D 0: 0 R 0 The coefficient of z vanishes if D1 p; hence D C exp 1 pdx , C a 2 2 constant, is the most general transformation with this property. Substitution into the 1 1 2 coefficient of z leads to r D2 4 p C q. 1.5. The first-order right factors of a L 2 type decomposition have the form 0 3 .1/ r l .C / D D p r C C ,wherep and r originate from the solution of the Riccati equation (notation as in (B.1)), and C is a constant. The Lclm for two operators of this form is 00 Á 00 r r L Á Lclm l.1/.C /; l.1/.C / D D2 C 2p D C p p0 C p2: 1 2 r0 r0 By division it is shown that any operator l.1/.C / is a divisor of L, i.e. it is contained in the left intersection ideal generated by it. A Solutions to the Exercises 183 L 2 1.6. For 1 , the first solution y1 is obtained from .D C a1/y D 0, the second from .D C a1/y DNy2 with yN2 a solution of .D C a2/y D 0. L 2 For 2 the two solutions are obtained from .D C ai /y D 0. Linear dependence over the base field would imply a relation q1y1 Cq2y2 D 0 withR q1;q2 from the base q1 field. Substituting the solutions this would entail Dexp .a2 a1/dx.Dueto q2 the non-equivalence of a1 and a2, its difference is not a logarithmic derivative; thus the right hand side cannot be rational. L 2 For 3 the equation .D Ca.C //y D 0 has to be solved, then C is specialized to N r0 r0 CN and CN . Substituting a1 D Cp and a2 D Cp in the above quotient, r C CN r C CN N q1 r C CN the integration may be performed with the result q D which is rational. p 2 r C CN 1.7. Define Á A2 4B. Two cases are distinguished. If ¤ 0,two 1 1 first-order right factors are l1;2 D D C 2 A ˙ 2 ; L has the decomposition L D Lclm.l ;l / of type L 2; a fundamental system is y D exp . 1 A ˙ 1 /x . 1 2 2 1;2 2 2 If D 0, the type L 2 decomposition is L D Lclm D C 1 A 1 , C a 3 2 x C C 1 1 constant; it yields the fundamental system y1 D exp 2 A , y2 D x exp 2 A .This result shows: A second-order lode with constant coefficients is always completely reducible. 1.8. According to Lemma 1.1, case .i/, the coefficient a of a first-order factor D C a has to satisfy 1 1 2 2 a00 3a01 C 1 a0 C a3 1 a2 C x a D 0: x x x x2 This second-order Riccati equation does not have a rational solution; thus a first- order right factor does not exist. According to case .ii/ ofthesamelemma,the coefficient b of a second-order factor D2 C bD C c follows from 2 2 4 2 2 2 b00 3b0b C 2 b0 C b3 2 b2 C x C 1 C b x D D 0: x x x x2 x x2 1 Its single rational solution b D 1 leads to c D x x and yields the second-order factor given in Example 1.3. 1.9. Let y1, y2,andy3 be a fundamental system for the homogeneous equation and W its Wronskian. The general solution may be written as Z r 0 0 y D C1y1 C C2y2 C C3y3 C y1 .y2y3 y2y3/dx Z W Z r r y .y y0 y0 y /dx C y .y y0 y0 y /dx 2 W 1 3 1 3 3 W 1 2 1 2 000 where C1, C2 and C3 are constants. For y D r with the fundamental system 2 y1 D 1, y2 D x and y3 D x there follows 184 A Solutions to the Exercises Z Z Z 1 y D C C C x C C x2 C 1 x2rdx x xrdx C x2 rdx: 1 2 3 2 2 1.10. A simple calculation shows that the commutator between l1 D D C a1 and 0 0 l2 D D C a2 vanishes if a1 a2 D 0,i.e.ifa1 and a2 differ by a constant. If this 2 0 is true, the representation (1.17) simplifies to L D D C .a1 C a2/D C a1a2 C a1; furthermore L D l1l2 D l2l1. An example of this case is D2 4xD C 4x2 3 D Lclm.D 2x C 1; D 2x 1/: An example of non-commutative first-order factors is 1 1 1 D2 1 C D C D Lclm D 1; D 1 C : x x x C 1 Chapter 2 2.1. The product of l1 Á @x C a1@y C b1 and l2 Á @x C a1@y C b2 is l1l2 D @xx C .a1 C a2/@xy C a1a2@yy C .b1 C b2/@x C.a2;x C a1a2;y C a1b2 C a2b1/@y C b2;x C a1b2;y C b1b2: The product l2l1 is obtained from this expression by interchange of all indices 1 and 2.
Load more

Recommended publications
  • Loewy Decomposition of Linear Differential Equations
    Bull. Math. Sci. (2013) 3:19–71 DOI 10.1007/s13373-012-0026-7 Loewy decomposition of linear differential equations Fritz Schwarz Received: 26 January 2012 / Revised: 11 June 2012 / Accepted: 27 June 2012 / Published online: 29 July 2012 © The Author(s) 2012. This article is published with open access at SpringerLink.com Abstract This paper explains the developments on factorization and decomposi- tion of linear differential equations in the last two decades. The results are applied for developing solution procedures for these differential equations. Although the subject is more than 100 years old, it has been rediscovered as recently as about 20 years ago. A fundamental ingredient has been the easy availability of symbolic computation systems to accomplish the extensive calculations usually involved in applications; to this end the interactive website http://www.alltypes.de has been provided. Although originally only developed for ordinary equations, it has been extended to large classes of partial equations as well. In the first part Loewy’s results for ordinary equations are outlined. Thereafter those results of differential algebra are summarized that are required for extending Loewy’s theory to partial equations. In the remaining part a fairly complete discussion of second- and some third-order partial differential equa- tions in the plane is given; it is shown that Loewy’s result remains essentially true for these equations. Finally, several open problems and possible extensions are discussed. Keywords Linear differential equations · Factorization · Loewy decomposition Mathematics Subject Classification (2000) Primary 54C40 · 14E20; Secondary 46E25 · 20C20 Communicated by N. Trudinger. F. Schwarz (B) Fraunhofer Gesellschaft, Institut SCAI, 53754 Sankt Augustin, Germany e-mail: [email protected] 123 20 F.
    [Show full text]
  • The Linear, Nonlinear and Partial Differential Equations Are Not Fractional Order Differential Equations
    Universal Journal of Engineering Science 3(3): 46-51, 2015 http://www.hrpub.org DOI: 10.13189/ujes.2015.030302 The Linear, Nonlinear and Partial Differential Equations are not Fractional Order Differential Equations Ali Karci Department of Computer Engineering, Inonu University, Turkey Copyright © 2015 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract The differential equations were considered as equations are satisfied by the current conditions; changing fractional order differential equations in literature. the conditions will yield new equations. Homotopy Analysis Method was used to obtain analytical The deficiencies in those methods in literature are coming solutions of these equations. We applied reverse processes to from the fractional order derivative definitions used in the analytical solutions of some fractional order differential many studies. The methods involving deficiencies yield equations, and observed that solutions could not satisfy the non-consistent results. Due to this case, there is a method corresponding equations. Due to this case, we proposed a proposed for fractional order derivative in Karcı studies new approach for fractional order derivative and it was [15,16] and by using this method concludes in whether verified by using this new approach that any differential converting any differential equations to fractional order equations cannot be converted into fractional order differential equations is valid or not. We saw that converting differential equations so simply. any differential equations into fractional order differential equations by changing order is not a valid method, since Keywords Derivatives, Fractional Calculus, Differential applications of the proposed method in Karcı’s papers Equations, Fractional Order Differential Equations illustrated that any differential equations cannot be converted MSC Classification: 26A33 into fractional order differential equations so simply.
    [Show full text]
  • Research Article Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra
    Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2016, Article ID 1796316, 9 pages http://dx.doi.org/10.1155/2016/1796316 Research Article Generation and Identification of Ordinary Differential Equations of Maximal Symmetry Algebra J. C. Ndogmo Department of Mathematics and Applied Mathematics, University of Venda, P/B X5050, Thohoyandou, Limpopo 0950, South Africa Correspondence should be addressed to J. C. Ndogmo; [email protected] Received 13 June 2016; Revised 25 October 2016; Accepted 7 November 2016 Academic Editor: Jaume Gine´ Copyright © 2016 J. C. Ndogmo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An effective method for generating linear ordinary differentialuations eq of maximal symmetry in their most general form is found, and an explicit expression for the point transformation reducing the equation to its canonical form is obtained. New expressions for the general solution are also found, as well as several identification and other results and a direct proof of the fact that a linear ordinary differential equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by an arbitrary function are also found, together with simple algebraic expressions for the corresponding general solution. 1. Introduction Computations with this algorithm remain however quite tedious and the authors managed to provide a general ex- Linear ordinary differential equations (lodes) are quite prob- pression for the lodes of maximal symmetry only up to the ablythemostcommontypeofdifferentialequationsthat order eight.
    [Show full text]
  • Decomposition of Ordinary Differential Equations
    Bull. Math. Sci. (2017) 7:575–613 https://doi.org/10.1007/s13373-017-0110-0 Decomposition of ordinary differential equations Fritz Schwarz1 Received: 26 May 2017 / Revised: 4 October 2017 / Accepted: 4 October 2017 / Published online: 22 November 2017 © The Author(s) 2017. This article is an open access publication Abstract Decompositions of linear ordinary differential equations (ode’s) into com- ponents of lower order have successfully been employed for determining their solutions. Here this approach is generalized to nonlinear ode’s. It is not based on the existence of Lie symmetries, in that it is a genuine extension of the usual solution algorithms. If an equation allows a Lie symmetry, the proposed decompositions are usually more efficient and often lead to simpler expressions for the solution. For the vast majority of equations without a Lie symmetry decomposition is the only avail- able systematic solution procedure. Criteria for the existence of diverse decomposition types and algorithms for applying them are discussed in detail and many examples are given. The collection of Kamke of solved equations, and a tremendeous compilation of random equations are applied as a benchmark test for comparison of various solution procedures. Extensions of these proceedings for more general types of ode’s and also partial differential equations are suggested. Keywords Ordinary differential equations · Decomposition · Exact solutions Mathematics Subject Classification Primary 34A05 · 34A34; Secondary 34C14 · 34B60 Communicated by Neil Trudinger. B Fritz Schwarz [email protected] 1 Fraunhofer Institute for Algorithms and Scientific Computing SCAI, 53754 Sankt Augustin, Germany 123 576 F. Schwarz 1 Introduction: description of the problem Ever since its introduction more than 300years ago the concept of a differential equa- tion, and connected to it finding its solutions, has been a fundamental problem in mathematics and its applications in natural sciences.
    [Show full text]