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SIAM J. on Mathematical Analysis

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1970

Volume 1, Issue 4, pp. 405-557


Lie Theory and Some Special Solutions of the Hypergeometric Equations

Willard Miller, Jr.

SIAM J. Math. Anal. 1, pp. 405-425 (21 pages) | Cited 1 time

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To obtain all solutions of the differential equations of hypergeometric type as basis functions corresponding to models of Lie algebra representations it is necessary to consider certain reducible representations. These representations are classified and models are constructed. A number of special hypergeometric functions arise naturally in the analysis, including the error and incomplete gamma functions, the incomplete beta functions, Legendre functions of the second kind, and some logarithmic solutions of the hypergeometric equation.

A Unitary Transform Related to Some Integral Equations

Kusum Soni

SIAM J. Math. Anal. 1, pp. 426-436 (11 pages)

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The integral equations \[ F(x) = \frac{d}{{dx}}\int_0^x {J_0 [2\sqrt {k(x - t)} ]f(t)dt} \] and \[ G(x) = \frac{d}{{dx}}\int_x^\infty {J_0 [2\sqrt {k(t - x)} ]g(t)dt} \] are usually considered separately, and their solutions \[ f(x) = \frac{d}{{dx}}\int_0^x {I_0 [2\sqrt {k(x - t)} ]F(t)dt} \] and \[ g(x) = \frac{d} {{dx}}\int_x^\infty {I_0 [2\sqrt {k(t - x)} ]G(t)dt} \] respectively are regarded unique. These solutions in general are not square integrable. We prove that under certain conditions either one of these two integral equations gives the square integrable solution of the other. Moreover, \[ \frac{d}{{dx}}\int_{ - \infty }^x {J_0 [2\sqrt {k(x - t)} ]f(t)dt} = 0 \qquad {\text{and}}\qquad \frac{d}{{dx}}\int_x^\infty {J_0 [2\sqrt {k(t - x)} ]g(t)} = 0,\quad - \infty < x < \infty , \] have nontrivial solutions. Therefore the assumption regarding the uniqueness of the solution of the second integral equation is not valid unless some additional conditions are specified.
It is well known that the homogeneous integral equation \[ \phi (x) = \int_{ - \infty }^\infty {k(x - y)\phi (y)dy} \] may have nontrivial solutions of the type $x^m e^{\beta x} $. The nontrivial solutions given here are not of this type.

Nonlinear Eigenvalue Problems for Some Fourth Order Equations. I: Maximal Solutions

Seymour V. Parter

SIAM J. Math. Anal. 1, pp. 437-457 (21 pages) | Cited 7 times

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A constructive, nonlinear iterative method is developed for the construction of a positive solution $(u(t),\theta (t))$ of nonlinear fourth order ordinary differential equations of the form $u'' = \lambda \theta H_1 (t,u,\theta ),\theta '' = \lambda uH_2 (t,u\theta )$. A solution $(u(t),\theta (t))$ is positive if $u(t) \leqq 0 \leqq \theta (t)$. Under appropriate hypothesis, these solutions are maximal in the sense that if $(\omega ,\Phi )$ is any other solution, then $u \leqq \omega $ and $\Phi \leqq \theta $. Thus, bounds on $(u,\theta )$ are a priori bounds on all solutions. Uniqueness is discussed. In special cases these positive solutions may be patched together to give other solutions.

Nonlinear Eigenvalue Problems for Some Fourth Order Equations. II: Fixed-Point Methods

Seymour V. Parter

SIAM J. Math. Anal. 1, pp. 458-478 (21 pages) | Cited 1 time

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Abstract Unavailable

On Boundary Value Problems for a Singularly Perturbed Differential Equation with a Turning Point

R. E. O’Malley, Jr.

SIAM J. Math. Anal. 1, pp. 479-490 (12 pages) | Cited 2 times

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A Representation of Linear Continuous Operators on Testing Functions and Distributions

Vaclav Dolezal

SIAM J. Math. Anal. 1, pp. 491-506 (16 pages) | Cited 7 times

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Abstract Unavailable

On the Asymptotic Behavior of Autonomous Differential Equations

R. H. Martin, Jr.

SIAM J. Math. Anal. 1, pp. 507-514 (8 pages)

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Abstract Unavailable

Noncontinuous Lyapunov Functions and Extendability of Solutions

Stephen R. Bernfeld

SIAM J. Math. Anal. 1, pp. 515-523 (9 pages)

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Abstract Unavailable

On the Reciprocal Modulus Relation for Elliptic Integrals

Henry E. Fettis

SIAM J. Math. Anal. 1, pp. 524-526 (3 pages)

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Abstract Unavailable

Inversion of a Convolution Transform Related to Heat Conduction

Harry Pollard and D. V. Widder

SIAM J. Math. Anal. 1, pp. 527-532 (6 pages) | Cited 2 times

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Abstract Unavailable

A Paradox in Asymptotics

F. W. J. Olver

SIAM J. Math. Anal. 1, pp. 533-534 (2 pages)

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An example is given of a convergent series expansion which has twice itself as its own asymptotic expansion.

The Nonoscillation of a Solution of a Third Order Equation

W. R. Utz

SIAM J. Math. Anal. 1, pp. 535-537 (3 pages)

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Abstract Unavailable

On the Evaluation of Certain Sums Involving the Natural Numbers Raised to an Arbitrary Power

Keith B. Oldham

SIAM J. Math. Anal. 1, pp. 538-546 (9 pages)

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Abstract Unavailable

Fractional Integrals of Distributions

A. Erdélyi and A. C. McBride

SIAM J. Math. Anal. 1, pp. 547-557 (11 pages) | Cited 4 times

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Certain operators of fractional integration arising in connection with singular differential operators, Hankel transforms, and dual integral equations involve integration of fractional order with respect to $r^2$ and multiplication of functions by fractional powers of the independent variable. Such operations are not meaningful for distributions. In this paper a class of generalized functions is introduced on which such operations can meaningfully be performed. The operations are defined as adjoints of corresponding operations on a suitably selected space of testing functions. Relations to spherically symmetric $n$-dimensional distributions and to the singular differential operator \[ \frac{{d^2 }}{{dr^2 }} + \frac{{2\nu + 1}}{r} + \frac{d}{{dr}} \] are discussed.
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