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About some Laplace's transforms and infinite power series

100%
Dymek F., Dymek J. F.
Mathematica Applicanda
|
1985
|
tom 13
|
nr 26
PL
.
EN
Many physical and technical problems are described by ordinary or partial differential eąuations. Among the analytical methods of solving differential eąuations, the methods based on the so-called integral transforms put forward on the first place [l6]. Possibility of making up the transforms tables of different functions is an advantage of integral methods of solving differential eąuations. Introducing into transforms [5 ] set some new elements is an aim of present elaboration. Theorem (1) appeared very helpful to transforms calculations. The authors also utilized the results of theorem (1) while sums of infinite power series calculation (2-13).
2
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On an integral transform by R. S. Phillips

51%
Bjon S.
Open Mathematics
|
2010
|
tom 8
|
nr 1
98-113
EN
The properties of a transformation $$ f \mapsto \tilde f_h $$ by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c-embedded space E is closed under the transformation. It is shown that $$ (\tilde f_h )\widetilde{_k } = \tilde f_{h + k} $$ for certain complex h and k, and that $$ f(t) = \lim _{h \to 0^ + } \tilde f_h (t) $$, where the limit is uniform in t on compact subsets of the positive real line. If f is Hölder-continuous at 0, then the limit is uniform on compact subsets of the non-negative real line. Inversion formulas for this transformation as well as for the Laplace transformation are derived. Transforms of certain semigroups of non-linear operators on a subset X of an L c-embedded space are studied through the C 0-semigroups, which they define by duality on a space of functions on X.
3
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A characterization of probability measures by f-moments

51%
Urbanik K.
Studia Mathematica
|
1996
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tom 118
|
nr 2
185-204
EN
Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments $ʃ_{0}^{∞} ƒ(x)μ^{*n}(dx)$ (n = 1,2...). A function ƒ is said to have the identification property} if probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function} if it is infinitely differentiable on the open half-line (0,∞) and $(-1)^{n} ƒ^{(n+1)}(x)$ is completely monotone for some nonnegative integer n. The purpose of this paper is to give a necessary and sufficient condition in terms of the representing measures for Bernstein functions to have the identification property.
4
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An efficient computational approach for linear and nonlinear fractional differential equations

51%
Singh J., Kumar D., Swroop R., Sharma R. P.
Waves, Wavelets and Fractals
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2017
|
tom 3
|
nr 1
1-13
EN
The pivotal aim of this article is to propose an efficient computational technique namely q-homotopy analysis transform method (q-HATM) to solve the linear and nonlinear time-fractional partial differential equation. In q-HATM iterative process, we investigate the behavior of independent variable for convergent series solution in admissible range. The q-HATM technique manipulates and controls the series solution, which rapidly converges to the exact solution in large admissible domain in a very efficient way. The solution procedure and explanation show the flexible efficiency of q-HATM, compared to other existing classical techniques for solving three different kind of time-fractional partial differential equations.
5
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A detailed analysis for the fundamental solution of fractional vibration equation

51%
Liu L.-L., Duan J.-S.
Open Mathematics
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2015
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tom 13
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nr 1
EN
In this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 < α < 1, while y(t) is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 < α < 2. We also consider the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y(t) for specified values of the coefficients and fractional order.
6
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Discrete Laplace cycles of period four

45%
Schröcker H.-P.
Open Mathematics
|
2012
|
tom 10
|
nr 2
426-439
EN
We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.
7
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Karhunen-Loève expansions of α-Wiener bridges

38%
Barczy M., Iglói E.
Open Mathematics
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2011
|
tom 9
|
nr 1
65-84
EN
We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation $$ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $$, with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t)t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function of the L 2[0, T]-norm square of X (α) studying also its asymptotic behavior (large and small deviation).
8
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Limiting distribution for a simple model of order book dynamics

38%
Kruk Ł.
Open Mathematics
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2012
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tom 10
|
nr 6
2283-2295
EN
A continuous-time model for the limit order book dynamics is considered. The set of outstanding limit orders is modeled as a pair of random counting measures and the limiting distribution of this pair of measure-valued processes is obtained under suitable conditions on the model parameters. The limiting behavior of the bid-ask spread and the midpoint of the bid-ask interval are also characterized.
9
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Tauberian theorems for vector-valued Fourier and Laplace transforms

38%
Chill R.
Studia Mathematica
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1998
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tom 128
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nr 1
55-69
EN
Let X be a Banach space and $f ∈ L^1_loc(ℝ;X)$ be absolutely regular (i.e. integrable when divided by some polynomial). If the distributional Fourier transform of f is locally integrable then f converges to 0 at infinity in some sense to be made precise. From this result we deduce some Tauberian theorems for Fourier and Laplace transforms, which can be improved if the underlying Banach space has the analytic Radon-Nikodym property.
10
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An asymptotic expansion for the distribution of the supremum of a random walk

32%
Sgibnev M. S.
Studia Mathematica
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2000
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tom 140
|
nr 1
41-55
EN
Let ${S_n}$ be a random walk drifting to -∞. We obtain an asymptotic expansion for the distribution of the supremum of ${S_n}$ which takes into account the influence of the roots of the equation $1-∫_ℝe^{sx}F(dx)=0,F$ being the underlying distribution. An estimate, of considerable generality, is given for the remainder term by means of submultiplicative weight functions. A similar problem for the stationary distribution of an oscillating random walk is also considered. The proofs rely on two general theorems for Laplace transforms.
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