Mutually conjugate solutions of formally self-adjoint differential equations

1979 ◽  
Vol 83 (1-2) ◽  
pp. 93-101 ◽  
Author(s):  
J. M. Hill ◽  
R. V. Nillsen
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Linear Differential Operator ◽  
Adjoint Operator ◽  
Linear Differential ◽  
Factorization Result

SynopsisLet L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n + 1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (−1)nQ*Q or L = (−1)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,−, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which yi and yn+j are mutually conjugate if i ≠ j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pólya.

Method of Forced Monotonicity for Conjugate type Boundary Value Problems for Ordinary Differential Equations

1988 ◽  
Vol 31 (1) ◽  
pp. 79-84
Author(s):  
P. W. Eloe ◽  
P. L. Saintignon
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Linear Differential Operator ◽  
Boundary Value ◽  
Linear Differential ◽  
Type Boundary ◽  
Sign Conditions ◽  
Conjugate Type

AbstractLet I = [a, b] ⊆ R and let L be an nth order linear differential operator defined on Cn(I). Let 2 ≦ k ≦ n and let a ≦ x1 < x2 < … < xn = b. A method of forced mono tonicity is used to construct monotone sequences that converge to solutions of the conjugate type boundary value problem (BVP) Ly = f(x, y),y(i-1) = rij where 1 ≦i ≦ mj, 1 ≦ j ≦ k, mj = n, and f : I X R → R is continuous. A comparison theorem is employed and the method requires that the Green's function of an associated BVP satisfies certain sign conditions.

scholarly journals New Approach for Solving Partial Differential Equations Based on Collocation Method

2020 ◽  
Vol 18 ◽  
pp. 118-128
Author(s):  
Alaa Almosawi ◽  
Luma N. M. Tawfiq
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Collocation Method ◽  
Linear Differential Operator ◽  
Partial Differential ◽  
Restrictive Assumption ◽  
New Approach ◽  
Linear Differential

In this paper, a new approach for solving partial differential equations was introduced. The collocation method based on LA-transform and proposed the solution as a power series that conforming Taylor series. The method attacks the problem in a direct way and in a straightforward fashion without using linearization, or any other restrictive assumption that may change the behavior of the equation under discussion. Five illustrated examples are introduced to clarifying the accuracy, ease implementation and efficiency of suggested method. The LA-transform was used to eliminate the linear differential operator in the differential equation.

scholarly journals Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Oksana Bihun ◽  
Clark Mourning
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Hermite Polynomials ◽  
Pseudospectral Method ◽  
Matrix Representations ◽  
The Family ◽  
Linear Differential

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family pνxν=0∞ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations Apν(x)=qν(x)pν(x), where A is a linear differential operator and each qν(x) is a polynomial of degree at most n0∈N; n0 does not depend on ν. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.

scholarly journals The factorization of the linear differential operator

2015 ◽  
Vol 63 (1) ◽  
pp. 139-151
Author(s):  
Klara Janglajew ◽  
Ewa Schmeidel
Keyword(s):  
Differential Operator ◽  
Linear Differential Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Factorization ◽  
Linear Differential ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Factorization Result

Abstract In this paper, necessary and sufficient conditions for factorization of a linear differential operator are presented. As a consequence of the factorization result some criterion of polynomial factorization is obtained. As a special case of the main result we have got Polynomial Remainder Theorem.

scholarly journals Approximate Solution of Intuitionistic Fuzzy Differential Equations with the Linear Differential Operator by the Homotopy Analysis Method

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
S. Melliani ◽  
Z. Belhallaj ◽  
M. Elomari ◽  
L. S. Chadli
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Homotopy Perturbation Method ◽  
Linear Differential Operator ◽  
Analysis Method ◽  
Intuitionistic Fuzzy ◽  
Homotopy Analysis ◽  
Linear Differential ◽  
Shed Light

In this work, the purpose is to discuss the homotopy analysis method (HAM) for the use of intuitionistic fuzzy differential equations with the linear differential operator. Furthermore, a numerical example is presented to shed light on the capability of the present method, and the numerical results illustrated by adopting the homotopy perturbation method (HPM) are compared with the exact solution to ensure the validity of our outcomes.

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