Theory Of Linear Differential Equations Research Articles

On the Applicability of a Certain Formula in the Theory of Linear Differential Equations

On the Applicability of a Certain Formula in the Theory of Linear Differential Equations

IV.—On an Elementary Solution of a Partial Differential Equation of Parabolic Type. Part I

SummaryProfessor E. T. Whittaker has recently discovered a Third Quantum-Mechanical Principal Function R(q, Q, t - T) and has worked out the theory of this function in detail when the Hamiltonian is By using the Sturm-Liouville theory of linear differential equations and the properties of Green's function, it is shown that the function is an elementary solution of the adjoint of the Schrodinger wave equation associated with the Hamiltonian H.It is pointed out that the modified Planck constant ħ arises solely from the commutation relation and may, from the analytical view-point, be any constant, real or complex. In particular, if ħ = i, the use of an algebra with the commutation relation leads to an elementary solution of the real equation of parabolic type

Singular point problems in the theory of linear differential equations

Singular point problems in the theory of linear differential equations

On Valiron's theory of linear differential equation of infinite order with constant coefficients

On Valiron's theory of linear differential equation of infinite order with constant coefficients

Book Review: Introduction to the Theory of Linear Differential Equations

Book Review: Introduction to the Theory of Linear Differential Equations

Analytic theory of linear differential equations

Analytic theory of linear differential equations

Adjoint differential equations

1. That adjoint differential equations have an analogue in the theory of linear difference equations seems to have been first observed by Bortolotti. The relation is essentially that of a matrix ǁ a rs ǁ to its transposed matrix ǁ a sr ǁ. It seems desirable, from this point of view, to carry out the transition from difference to differential equations, and thus prove that the analogy is a real one. This is done in Art. 2. There are further consequences of general interest. A set of linear equations corresponds to a differential equation and its boundary conditions, and thus we can find an interpretation of the adjoint boundary conditions introduced by Birkhoff into the theory of linear differential equations (Arts. 3-6). The relation between the two Green’s functions, implicit in Birkhoff’s work, then becomes evident (Art. 7). 2. We first prove that if the equations a r 1 y 1 + a r 2 y 2 + ... + a rn y n = fr ( r = 1 to n ) (1) are so constituted that they merge into the differential equation L ( y ) Ξ a m d m y / dx m . . . + a 1 dy / dx + a o y = f (2) by passing to an infinite number of infinitesimally spaced unknowns, the transposed equations a 1 r z 1 + a 2 r z 2 + ... + a nr z n = g r (3) merge into the adjoint equation M ( z ) Ξ (—) m d m / dx m ( a m z ) + ... - d / dx ( a 1 z ) + a o z = g .(4)

The differential equation of the elliptic cylinder

It is not capable of reference to the hypergeometric type and so requires methods peculiar to itself. The principal difficulty in the solution of the equation consists in the determination of a number, the characteristic exponent, which appears in the solution. It is known from general considerations of the theory of linear differential equations that for the trigonometric form of the equation, y may be expressed as

An unusual type of expansion problem

In the majority of papers on the expansion problems associated with linear differential systems or with integral equations, the chief aim is a proof that a function, arbitrary within the limits of certain restrictions as light as possible, can be expanded in a uniformly convergent series of functions arising from the differential system or the integral equation. As exceptions to the rule we may cite a number of papers dealing with differential systems for which the boundary conditions are of irregular type.t In the following paragraphs we present a study of the linear differential system of the first order, without the customary limitation that a certain coefficient in the differential equation remain positive; the results obtained are in sharp contrast not only with those found for the usual expansion problems and those already mentioned in connection with irregular boundary conditions, but also with the facts for the precisely analogous second order differential system.+ A second point of some interest is the appearance of a normal orthogonal set of trigonometric functions differing from the normalized Fourier set but as closely related to it as are the sine and cosine sets. It is interesting that this exceedingly simple differential system seems to have escaped attention entirely. So far as we know, the existence of this gap in the theory of boundary value and expansion problems was first pointed out by Professor Birkhoff in a recent course of lectures on the theory of linear differential equations.

Societies and Academies

PARIS. Academy of Sciences, September 13.—M. Léon Guignard in the chair.—F. E. Fournier: The apparent displacement of some stars in the total eclipse of the sun of May 29, 1919.—A. Blondel: The calculation of electric cables by the use of vectorial functions with real notation. The method described has the same advantages as when imaginary quantities are employed, but only real quantities are utilised in the demonstration. It is based on the introduction of vectorial series.—V. Smirnoff: Some points of the theory of linear differential equations of the second order and automorphic functions.—E. Jouguet: The velocity of waves in elastic solids.—C. Gamichel: The transmission of energy by the vibrations of water in pipes. Remarks on some recent publications of M. Cons tan tinescu, and a statement of the work done by the author on the same subject.—E. Canals: The estimation of calcium and magnesium in different saline media. A study of the conditions under which, in acetic acid solutions, it is possible to separate completely calcium and magnesium from salts of iron and aluminium.—G. Zeil: The rôle of building corals in lithospheric re-adjustments.—C. Störmer: Some rays of aurora observed on March 22, 1920, which reached a height of 500 km. The aurora borealis of March 22 was photographed from seven stations under favourable conditions. The stations were connected by telephone, and simultaneous photographs were taken from two or three stations at a time. About 620 photographs were obtained, and they show that the summits of some of the rays reached an altitude of 500 km. above the earth.—A. Chevalier: The origin of the cider apple-trees cultivated in Normandy and Brittany.—F. Viès: The spectral properties of the tetanus toxin. Spectrophotometric studies of the ultraviolet absorption spectra of the effects of heating and of the addition of antitoxin to solutions of the tetanus toxin.—A. Marie and L. MacAuliffe: The influence of life in Paris on the race. A study of 1509 Parisians of the poorer classes, 850 of whom were born of provincial parents, 294 of Parisian parents, and the remainder of one Parisian and one provincial parent. The Paris climate and town life lead to modifications which are thus summarised: The hair and eyes less pigmented than in the rest of France, more marked cranial development in proportion to height, and shortening of the limbs.—J. L. Dantan: Budding in Antipathella subpinnata and Parantipathes larix.

On a theory of linear differential equations in general analysis

The fundamental principles of generalization have been well formulated by E. H. Moore in the statement: The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features. In applying this principle to the analogies between the theory of linear integral equations and the theory of systems of linear algebraic equations and biquadratic forms, Moore was led to the construction of a body of General Analysis, consisting of a theory of properties of classes of functions on a general range, developed in his Introduction to general analysis;t a theory of functional operations, sketched in the papers, On a form of general analysis with applications to linear differential and integral equations, t and On the fundamental functional operation of a general theory of linear integral equations;? a general theory of linear integral equations, outlined in the paper on Foundations of the theory of linear integral equations|| and the paper on Functional operations; and an existence theorem for general differential equations. ? It stands to reason that these general theories, in addition to the wide range of results obtainable as a result of specialization, may also serve as a basis for the consideration of generalizations in analogous fields. The present paper is an application of some of the concepts of the General Analysis and some of the results of the general theory of integral equations to the theory of general linear differential equations. The possibility of such an application

On a Theory of Linear Differential Equations in General Analysis

On a Theory of Linear Differential Equations in General Analysis

Note on asymptotic expressions in the theory of linear differential equations

Note on asymptotic expressions in the theory of linear differential equations

Errata: “On a theory of linear differential equations in general analysis” [Trans. Amer. Math. Soc. 18 (1917), no. 1, 73–96; 1501063

Errata: “On a theory of linear differential equations in general analysis” [Trans. Amer. Math. Soc. 18 (1917), no. 1, 73–96; 1501063

On Certain Asymptotic Expressions in the Theory of Linear Differential Equations

On Certain Asymptotic Expressions in the Theory of Linear Differential Equations

The Elliptic Cylinder Functions of the Second Kind

The differential equation of Mathieu, or the equation of the elliptic cylinder functionsis known by the theory of linear differential equations to have a general solution of the typeφ and ψ being periodic functions of z, with period 2π.

On the Theory of Linear Difference Equations

On the Theory of Linear Difference Equations

On the general solution of Mathieu's equation

The differential equation of Mathieu, or “equation of the elliptic cylinder functions,”occurs in many physical and astronomical problems. From the general theory of linear differential equations, we learn that its solution is of the typewhere A and B denote arbitrary constants, μ is a constant depending on the constants a and q of the differential equation, and φ(z) and ψ(z) are periodic functions of z. For certain values of a and q the constant μ vanishes, and the solution y is then a purely periodic function of z; but in general μ is different from zero.

Poznámka k theorii rovnic differenciálních lineárních. [IV.

Poznámka k theorii rovnic differenciálních lineárních. [IV.

Poznámka k theorii rovnic differenciálních lineárních. [III.

Poznámka k theorii rovnic differenciálních lineárních. [III.