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# Integral Transforms in Applied Mathematics

30 Nov 1971-

TL;DR: In this paper, an intermediate-level text on the use of integral transforms in applied mathematics and engineering is presented, which is divided into five parts covering integral transform pairs, the Laplace transform, Fourier transforms, Hankel transform, and finite Fourier transform.

Abstract: An intermediate-level text on the use of integral transforms in applied mathematics and engineering. Existing works either cover the subject in more elementary form or are advanced treatises. In a very lucid style the author deals with the use of this important mathematical tool to solve ordinary and partial differential equations in problems in electrical circuits, mechanical vibration and wave motion, heat conduction, and fluid mechanics. The book is divided into five parts covering integral transform pairs, the Laplace transform, Fourier transforms, Hankel transforms, and finite Fourier transforms. A basic knowledge of complex variables and elementary differential equations is assumed. There are many exercises and examples drawn from the above fields, tables of the transform pairs needed in the text, and a glossary of terms with which the student may be unfamiliar. For the student who seeks further background on the subject, an annotated bibliography is provided.

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TL;DR: This paper presents a preliminary study on how to review solar irradiance and photovoltaic power forecasting using text mining, which serves as the first part of a forthcoming series of text mining applications in solar forecasting.

Abstract: Text mining is an emerging topic that advances the review of academic literature. This paper presents a preliminary study on how to review solar irradiance and photovoltaic (PV) power forecasting (both topics combined as “solar forecasting” for short) using text mining, which serves as the first part of a forthcoming series of text mining applications in solar forecasting. This study contains three main contributions: (1) establishing the technological infrastructure (authors, journals & conferences, publications, and organizations) of solar forecasting via the top 1000 papers returned by a Google Scholar search; (2) consolidating the frequently-used abbreviations in solar forecasting by mining the full texts of 249 ScienceDirect publications; and (3) identifying key innovations in recent advances in solar forecasting (e.g., shadow camera, forecast reconciliation). As most of the steps involved in the above analysis are automated via an application programming interface, the presented method can be transferred to other solar engineering topics, or any other scientific domain, by means of changing the search word. The authors acknowledge that text mining, at its present stage, serves as a complement to, but not a replacement of, conventional review papers.

348 citations

### Cites background from "Integral Transforms in Applied Math..."

...The main motivation for a transformation is that the determination or manipulation of F(p) is often more convenient than that of f (x) (Miles, 1971)....

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TL;DR: In this paper, the Sumudu transform of partial derivatives is derived, and its applicability demonstrated using three different partial differential equations (PDEs) is demonstrated with respect to three different PDEs.

Abstract: The Sumudu transform of partial derivatives is derived, and its applicability demonstrated using three different partial differential equations.

114 citations

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TL;DR: In this paper, a linearized system of equations for the atmosphere's first internal mode in the vertical is derived, which governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height.

Abstract: A linearized system of equations for the atmosphere's first internal mode in the vertical is derived. The system governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height. When the basic flow is at rest, solutions for the transient and final adjusted state are found by the method of Hankel transforms. Two examples are considered, one with an initial top hat potential vorticity and one with an initial Gaussian-type potential vorticity. These two examples, which extend the work of Fischer (1963) and Obukhov (1949), indicate that the energetical efficiency of cloud-cluster-scale heating in producing balanced vortex flow is very low, on the order of a few percent. The vast majority of the energy is simply partitioned to gravity-inertia waves. In contrast the efficiency of cloud-cluster-scale vorticity transport is very high. When the basic state possesses positive relative vorticity in an inner region, the energy partition can be subst...

108 citations

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TL;DR: In this paper, the diffraction of a surface wave that is obliquely incident upon a small cylindrical deformation of the bottom of a laterally unbounded ocean is calculated by small perturbation theory.

Abstract: The diffraction of a surface wave that is obliquely incident upon a small, cylindrical deformation of the bottom of a laterally unbounded ocean is calculated by small perturbation theory. The reflection coefficient is found to vanish for an angle of incidence of 45° independently of the shape of the obstacle.

79 citations

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TL;DR: In this paper, a framework for estimating aquifer hydraulic properties using sinusoidal pumping is presented that derives analytical solutions for confined, leaky, and partially penetrating conditions, and compares the analytical solutions with a finite element model.

Abstract: A framework for estimating aquifer hydraulic properties using sinusoidal pumping is presented that (1) derives analytical solutions for confined, leaky, and partially penetrating conditions; (2) compares the analytical solutions with a finite element model; (3) establishes a field protocol for conducting sinusoidal aquifer tests; and (4) estimates aquifer parameters using the analytical solutions. The procedure is demonstrated in one surficial and two confined aquifers containing potentially contaminated water in coastal plain sediments at the Savannah River site, a federal nuclear facility. The analytical solutions compare favorably with finite-element solutions, except immediately adjacent to the pumping well where the assumption of zero borehole radius is not valid. Estimated aquifer properties are consistent with previous studies for the two confined aquifers, but are inconsistent for the surficial aquifer; conventional tests yielded estimates of the specific yield—consistent with an unconfined response—while the shorter-duration sinusoidal perturbations yielded estimates of the storativity—consistent with a confined, elastic response. The approach minimizes investigation-derived wastes, a significant concern where contaminated fluids must be disposed of in an environmentally acceptable manner. An additional advantage is the ability to introduce a signal different from background perturbations, thus easing detection.

73 citations

### Cites background from "Integral Transforms in Applied Math..."

...The transform has the property that: Fc f 00ðr; z; tÞf g ¼ np b 2 fcðr; n; tÞ þ ð 1Þnf 0 ðr;m; tÞ f 0ðr; 0; tÞ ð61Þ where the prime denotes differentiation with respect to z (Miles 1971; Pinkus and Zafrany 1977; Sneddon 1972)....

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...f 0ðr; 0; tÞ ð61Þ where the prime denotes differentiation with respect to z (Miles 1971; Pinkus and Zafrany 1977; Sneddon 1972)....

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