Fixed points of differential polynomials generated by solutions of complex linear differential equations

I. Third Order Linear Homogeneous Differential Equations in Normal Form.- 1. Fundamental Properties of Solutions of the Third Order Linear Homogeneous Differential Equation.- 1. The Normal Form of a Third Order Linear Homogeneous Differential Equation.- 2. Adjoint and Self-adjoint Third Order Linear Differential Equations.- 3. Fundamental Properties of Solutions.- 4. Relationship between Solutions of the Differential Equations (a) and (b).- 5. Integral Identities.- 6. Notion of a Band of Solutions of the First, Second and Third Kinds.- 7. Further Properties of Solutions of the Differential Equation (a) Implied by Properties of Bands.- 8. Weakening of Property (v) for the Laguerre Invariant.- 2. Oscillatory Properties of Solutions of the Differential Equation (a).- 1. Basic Definitions.- 2. Sufficient Conditions for the Differential Equation (a) to Be Disconjugate.- 3. Sufficient Conditions for Oscillatoricity of Solutions of the Differencial Equation (a).- 4. Further Conditions Concerning Oscillatoricity or Non-oscillatoricity of Solutions of the Differential Equation (a).- 5. Relation between Solutions without Zeros and Oscillatoricity of the Differential Equation (a).- 6. Sufficient Conditions for Oscillatoricity of Solutions of the Differential Equation (a) in the Case A(x) ? 0, x ? (a, ?).- 7. Conjugate Points, Principal Solutions and the Relationship between the Adjoint Differential Equations (a) and (b).- 8. Criteria for Oscillatoricity of the Differential Equations (a) and (b) Implied by Properties of Conjugate Points.- 9. Further Criteria for Oscillatoricity of the Differential Equation (b).- 10. The Number of Oscillatory Solutions in a Fundamental System of Solutions of the Differential Equation (a).- 11. Criteria for Oscillatoricity of Solutions of the Differential Equation (a) in the Case that the Laguerre Invariant Does Not Satisfy Condition (v).- 12. The Case, When the Laguerre Invariant Is an Oscillatory Function of x.- 13. The Differential Equation (a) Having All Solutions Oscillatory in a Given Interval.- 3. Asymptotic Properties of Solutions of the Differential Equations (a) and (b).- 1. Asymptotic Properties of Solutions without Zeros of the Differential Equations (a) and (b).- 2. Asymptotic Properties of Oscillatory Solutions of the Differential Equation (b).- 3. Asymptotic Properties of All Solutions of the Differential Equation (a).- 4. Boundary Value Problems.- 1. The Green Function and Its Applications.- 2. Further Applications of Integral Equations to the Solution of Boundary-value Problems.- 3. Generalized Sturm Theory for Third Order Boundary-value Problems.- 4. Special Boundary-value Problems.- II. Third Order Linear Homogeneous Differential Equations with Continuous Coefficients.- 5. Principal Properties of Solutions of Linear Homogeneous Third Order Differential Equations with Continuous Coefficients.- 1. Principal Properties of Solutions of the Differential Equation (A).- 2. Bands of Solutions of the Differential Equation (A).- 3. Application of Bands to Solving a Three-point Boundary-value Problem.- 6. Conditions for Disconjugateness, Non-oscillatoricity and Oscillatoricity of Solutions of the Differential Equation (A).- 1. Conditions for Disconjugateness of Solutions of the Differential Equation (A).- 2. Solutions without Zeros and Their Relation to Oscillatoricity of Solutions of the Differential Equation (A).- 3. Conditions for the Existence of Oscillatory Solutions of the Differential Equation (A).- 4. On Uniqueness of Solutions without Zeros of the Differential Equation (A).- 5. Some Properties of Solutions of the Differential Equation (A) with r(x) ? 0.- 7. Comparison Theorems for Differential Equations of Type (A) and Their Applications.- 1. Comparison Theorems.- 2. A Simple Application of Comparison Theorems.- 3. Remark on Asymptotic Properties of Solutions of the Differential Equation (A).- III. Concluding Remarks.- 1. Special Forms of Third Order Differential Equations.- 2. Remark on Mutual Transformation of Solutions of Third Order Differential Equations.- IV. Applications of Third Order Linear Differential Equation Theory.- 8. Some Applications of Linear Third Order Differential Equation Theory to Non-linear Third Order Problems.- 1. Application of Quasi-linearization to Certain Problems Involving Ordinary Third Order Differential Equations.- 2. Three-point Boundary-value Problems for Third Order Non-linear Ordinary Differential Equations.- 3. On Properties of Solutions of a Certain Non-linear Third Order Differential Equation.- 9. Physical and Engineering Applications of Third Order Differential Equations.- 1. On Deflection of a Curved Beam.- 2. Three-layer Beam.- 3. Survey of Some Other Applications of Third Order Differential Equations.- References.

1. The basic idea of the application of integral operators to the Weierstrass-Hadamard direction. In order to generate and investigate solutions of differential equations, operators p (defined as the integral operators of the first kind) have been introduced in [2; 6](2). p transforms analytic functions of one and two variables into solutions of linear elliptic differential equations of two and three variables, respectively. It has been shown in the abovementioned papers that p (as well as some other operators connected with p) preserves many properties of the functions to which the operator is applied. This situation permits us to use theorems in the theory of functions to obtain theorems not merely on harmonic functions in two variables, but on solutions of other linear differential equations as well(3). In the present paper the above-mentioned method is used to prove connections between the properties in the large of solutions i1 of certain linear differential equations, see (1.1) and (1.3), on one side and the structure of certain subsequences of the coefficients of the series development of V/ at the origin on the other. Let us formulate these procedures in a somewhat more concrete manner, at first for equations in two variables. Let i1 be a (real) solution of the differential equation

The purpose of this study is to give a Taylor polynomial approximation for the solution of hyperbolic type partial differential equations with constant coefficients. The technique used is an improved Taylor matrix method, which has been given for solving ordinary differential, integral and integro-differential equations [M. Gülsu and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82(5) (2005), pp. 629–642; M. Gülsu and M. Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. Math. Comput. 188 (2007), pp. 446–449; A. Karamete and M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 79(9) (2002), pp. 987–1000; N. Kurt and M. Çevik, Polynomial solution of the single degree of freedom system by Taylor matrix method, Mech. Res. Commun. 35 (2008), pp. 530–536; N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Inst. 345 (2008), pp. 839–850; Ş. Nas, S. Yalçinbaş, and M. Sezer, A method for approximate solution of the high-order linear Fredholm integro-differential equations, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, Taylor polynomial solution of Volterra integral equations, Int. J. Math. Edu. Sci. Technol. 25(5) (1994), pp. 625–633; M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, M. Gülsu, and B. Tanay, A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials, J. Franklin Inst. 343 (2006), pp. 647–659; S. Yalçinbaş, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equation, Appl. Math. Comput. 127 (2002), pp. 196–206; S. Yalçinbaş and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000), pp. 291–308]. Some numerical examples, which consist of initial and boundary conditions, are given to illustrate the reliability and efficiency of the method. Also, the results obtained are compared by the known results; the error analysis is performed and the accuracy of the solution is shown.

4 - Higher-Order Differential Equations

Previous article Next article Full AccessConditions for the Existence of Conjugate Points for a Fourth Order Linear Differential EquationJohn S. BradleyJohn S. Bradleyhttps://doi.org/10.1137/0117086PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Dallas Banks, Bounds for the eigenvalues of some vibrating systems, Pacific J. Math., 10 (1960), 439–474 MR0117378 0097.16501 CrossrefGoogle Scholar[2] John H. Barrett, Two-point boundary problems for linear self-adjoint differential equations of the fourth order with middle term, Duke Math. J., 29 (1962), 543–554 10.1215/S0012-7094-62-02955-1 MR0148981 0108.08204 CrossrefISIGoogle Scholar[3] E. S. Čičkin, A non-oscillation theorem for a linear self-adjoint differential equation of fourth order, Izv. Vysš. Učebn. Zaved. Matematika, 1960 (1960), 206–209 MR0137900 Google Scholar[4] A. M. Fink, The functional T$\int \sb{0}\sp{T}$R and the zeroes of a second order linear differential equation, J. Math. Pures Appl. (9), 45 (1966), 387–394 MR0208053 0144.09402 Google Scholar[5] A. M. Fink, On the zeros of $y\sp{\prime\prime}+py=0$ with linear, convex and concave p, J. Math. Pures Appl. (9), 46 (1967), 1–10 MR0213651 0153.11401 Google Scholar[6] Don B. Hinton, Clamped end boundary conditions for fourth-order selfadjoint differential equations, Duke Math. J., 34 (1967), 131–138 10.1215/S0012-7094-67-03415-1 MR0208054 0148.06402 CrossrefISIGoogle Scholar[7] Walter Leighton, On the zeros of solutions of a second-order linear differential equation, J. Math. Pures Appl. (9), 44 (1965), 297–310 MR0186868 0128.30803 Google Scholar[8] Walter Leighton, Erratum: “On the zero of solutions of a second-order linear differential equation”, J. Math. Pures Appl. (9), 46 (1967), 10– MR0214856 0189.37401 Google Scholar[9] Walter Leighton and , Zeev Nehari, On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc., 89 (1958), 325–377 MR0102639 0084.08104 CrossrefGoogle Scholar[10] A. Ju. Levin, Distribution of the zeros of solutions of a linear differential equation, Soviet Math. Dokl., 5 (1964), 818–821 0117.05003 Google Scholar[11] L. D. Nikolenko, Some criteria for non-oscillation of a fourth order differential equation, Dokl. Akad. Nauk SSSR (N.S.), 114 (1957), 483–485 MR0091394 0079.11101 Google Scholar[12] William T. Reid, Riccati matrix differential equations and non-oscillation criteria for associated linear differential systems, Pacific J. Math., 13 (1963), 665–685 MR0155049 0119.07401 CrossrefISIGoogle Scholar[13] Thomas L. Sherman, Properties of solutions of $n{\rm th}$ order linear differential equations, Pacific J. Math., 15 (1965), 1045–1060 MR0185185 0132.31204 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Lyapunov-Type Inequalities for Higher-Order Linear Differential EquationsLyapunov Inequalities and Applications | 28 January 2021 Cross Ref Existence of conjugate points for second and fourth order differential equationsProceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 89, No. 3-4 | 14 November 2011 Cross Ref The existence of conjugate points for selfadjoint differential equations of even orderProceedings of the American Mathematical Society, Vol. 56, No. 1 | 1 January 1976 Cross Ref Green's function for n-n boundary value problem and an analogue of Hartman's resultJournal of Mathematical Analysis and Applications, Vol. 51, No. 3 | 1 Sep 1975 Cross Ref Separation Theorems for Self-Adjoint Linear Differential Equations of the Fourth OrderWilliam Brunner MillerSIAM Journal on Mathematical Analysis, Vol. 6, No. 4 | 17 February 2012AbstractPDF (1582 KB)Bounds for Eigenvalues and Conditions for Existence of Conjugate PointsD. O. BanksSIAM Journal on Applied Mathematics, Vol. 27, No. 3 | 12 July 2006AbstractPDF (1060 KB)Oscillation Criteria for a Class of Fourth Order Differential EquationsKurt KreithSIAM Journal on Applied Mathematics, Vol. 22, No. 1 | 12 July 2006AbstractPDF (189 KB) Volume 17, Issue 5| 1969SIAM Journal on Applied Mathematics835-1015 History Submitted:05 January 1968Published online:12 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0117086Article page range:pp. 984-991ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

One of the effective methods to find explicit solutions of differential equations is the method based on the operator representation of solutions. The essence of this method is to construct a series, whose members are the relevant iteration operators acting to some classes of sufficiently smooth functions. This method is widely used in the works of B. Bondarenko for construction of solutions of differential equations of integer order. In this paper, the operator method is applied to construct solutions of linear differential equations with constant coefficients and with Caputo fractional derivatives. Then the fundamental solutions are used to obtain the unique solution of the Cauchy problem, where the initial conditions are given in terms of the unknown function and its derivatives of integer order. Comparison is made with the use of Mikusinski operational calculus for solving similar problems.

One of the effective methods for finding exact solutions of differential equations is the method based on the operator representation of solutions. The essence of this method is to construct a series, whose members are the relevant iteration operators acting to some classes of sufficiently smooth functions. This method is widely used in the papers of Bondarenko for construction of solutions of differential equations of the integer order. In this paper, the operator method is applied to construct solutions of linear differential equations with constant coefficients and generalized Riemann-Liouville fractional derivative of order α and type γ. Then fundamental solutions are used to obtain the unique solution of the Cauchy problem.

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

In this paper, we study the growth of solutions of the second order linear complex differential equations insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

A symbolic algorithm for exact power series solutions of nth order linear homogeneous differential equations with polynomial coefficients near an ordinary point

The solution of linear algebraic equations is probably the most important topic in numerical methods. Since the simplest models for the physical world are linear, linear equations arise frequently in physical problems. Even the most complicated situations are frequently approximated by a linear model as a first step. Further, as will be seen in Chapter 7, the solution of a system of nonlinear equations is achieved by an iterative procedure involving the solution of a series of linear systems, each of them approximating the nonlinear equations. Similarly, the solution of differential and integral equations using finite difference method leads to a system of linear or nonlinear equations. Linear equations also arise frequently in numerical analysis. For example, the method of undetermined coefficients which is useful for deriving formulae for numerical differentiation, integration or solution of differential equations, generally leads to a system of linear equations.

In this paper, we have found the solution of linear sequential fractional differential equations involving conformable fractional derivatives of order with constant coefficients. For this purpose, we first discussed fundamental properties of the conformable derivative and then obtained successive conformable derivatives of the fractional exponential function. After this, we determined the analytic solution of linear sequential fractional differential equations (L.S.F.D.E.) in terms of a fractional exponential function. We have demonstrated this developed method with a few examples of homogeneous linear fractional differential equations. This method gives a conjugation with the method to solve classical linear differential equations with constant coefficients.

Next article Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential EquationsStanisław LewanowiczStanisław Lewanowiczhttps://doi.org/10.1137/0517074PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA method is presented for obtaining recurrence relations for the coefficients in Jacobi series solutions of linear ordinary differential equations with polynomial coefficients.[1] C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc., 53 (1957), 134–149 18,516a 0077.32503 CrossrefGoogle Scholar[2] David Elliott, The expansion of functions in ultraspherical polynomials, J. Austral. Math. Soc., 1 (1959/1960), 428–438 23:A1997 0099.28603 CrossrefGoogle Scholar[3] A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953 Google Scholar[4] L. Fox, Chebyshev methods for ordinary differential equations, Comput. 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Mat., 17 (1983), 655–675 85d:33030 0591.65089 Google Scholar[10] Stanisław Lewanowicz, Recurrence relations for hypergeometric functions of unit argument, Math. Comp., 45 (1985), 521–535 86m:33004 0583.33005 CrossrefISIGoogle Scholar[11] Y. L. Luke, The Special Functions and their Approximations, Academic Press, New York, 1969 Google Scholar[12] A. Magnus, Application des récurrences au calcul d'une classe d'intégrales, Rep., 71, Inst. Math. Pure Appl., Univ. de Louvain, 1974 Google Scholar[13] A. G. Morris and , T. S. Horner, Chebyshev polynomials in the numerical solution of differential equations, Math. Comp., 31 (1977), 881–891 56:1729 0386.65040 CrossrefISIGoogle Scholar[14] O. Oluremi Olaofe, On the Tchebyschev method of solution of ordinary differential equations, J. Math. Anal. Appl., 60 (1977), 1–7 10.1016/0022-247X(77)90043-9 56:1724 0363.65065 CrossrefISIGoogle Scholar[15] Stefan Paszkowski, Zastosowania numeryczne wielomianów i szeregów Czebyszewa, Państwowe Wydawnictwo Naukowe, Warsaw, 1975, 481– 56:13534 0423.65012 Google Scholar[16] N. Robertson, An ALTRAN program for finding a recursion formula for the Gegenbauer coefficients of a function, Spec. Rep., SWISK 11, Nat. Res. Inst. for Math. Sci., Pretoria, 1979 Google Scholar[17] Jet Wimp, Computation with recurrence relations, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984xii+310 85f:65001 Google ScholarKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equation Next article FiguresRelatedReferencesCited byDetails Descriptions of fractional coefficients of Jacobi polynomial expansions18 April 2022 | The Journal of Analysis, Vol. 30, No. 4 Cross Ref On Jacobi polynomials and fractional spectral functions on compact symmetric spaces4 January 2021 | The Journal of Analysis, Vol. 29, No. 3 Cross Ref Spectral Solutions for Differential and Integral Equations with Varying Coefficients Using Classical Orthogonal Polynomials17 July 2018 | Bulletin of the Iranian Mathematical Society, Vol. 45, No. 2 Cross Ref On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kindsActa Mathematica Scientia, Vol. 35, No. 2 Cross Ref On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials6 January 2004 | Journal of Physics A: Mathematical and General, Vol. 37, No. 3 Cross Ref On the coefficients of differentiated expansions and derivatives of Jacobi polynomials8 April 2002 | Journal of Physics A: Mathematical and General, Vol. 35, No. 15 Cross Ref The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable functionJournal of Computational and Applied Mathematics, Vol. 89, No. 1 Cross Ref On the legendre coefficients of the moments of the general order derivative of an infinitely differentiable functionInternational Journal of Computer Mathematics, Vol. 56, No. 1-2 Cross Ref Evaluation of Bessel function integrals with algebraic singularitiesJournal of Computational and Applied Mathematics, Vol. 37, No. 1-3 Cross Ref Properties of the polynomials associated with the Jacobi polynomials1 January 1986 | Mathematics of Computation, Vol. 47, No. 176 Cross Ref Volume 17, Issue 5| 1986SIAM Journal on Mathematical Analysis History Submitted:15 April 1985Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordsJacobi seriesJacobi coefficientsrecurrence relationsdifference operatorslinear differential equationMSC codes42C1039A7065L0565L10PDF Download Article & Publication DataArticle DOI:10.1137/0517074Article page range:pp. 1037-1052ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order

The major goal of this paper is to find accurate solutions for linear fractional differential equations of order . Hence, it is necessary to carry out this goal by preparing a new method called Fractional Finite Difference Method (FFDM). However, this method depends on several important topics and definitions such as Caputo’s definition as a definition of fractional derivative, Finite Difference Formulas in three types (Forward, Central and Backward) for approximating the and derivatives and Composite Trapezoidal Rule for approximating the integral term in the Caputo’s definition. In this paper, the numerical solutions of linear fractional differential equations using FFDM will be discussed and illustrated. The purposed problem is to construct a method to find accurate approximate solutions for linear fractional differential equations. The efficiency of FFDM will be illustrated by solving some examples of linear fractional differential equations of order .