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The new (DαG)/G expansion method to solve fractional KdV-equations

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arXiv 2017年

作者： Ghosh, Uttam Sarkar, Susmita Das, Shantanu Department of Applied Mathematics University of Calcutta Kolkata India Reactor Control Systems Design Section e and i Group Barc Mumbai

Fractional calculus of variation plays an important role to formulate the non-conservative physical problems. in this paper we use semi-inverse method and fractional variational principle to formulate the fractional order generalized Korteweg-de Vries (KdV) equation with Jumarie type fractional derivative and proposed a new method to solve the non-linear fractional differential equation named as DαG/G expansion method. Using this method we obtained the solutions of fractional order generalized KdV. The obtained solutions are more general compare to other method and the solutions are expressed in terms of the generalized hyperbolic, trigonometric functions and rational functions. Copyright © 2017, The Authors. All rights reserved.

关键词： Calculations

Characterization of left ventricular hypertrophy via fractional derivatives

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arXiv 2017年

in this paper we have used the concepts of fractional derivative of rough curves to characterize eCG of LVH patients and compared the results with normal eCGs. in mathematical language an eCG is a rough curve having Q,R,S points as non differentiable points where classical derivatives do not exist but fractional derivatives exist. We have calculated both left and right modified Riemann-Liouville fractional derivatives and their differences termed as phase transition at those non differentiable points of V1, V2, V5 and V6 leads. investigation shows that phase transition is higher for LVH patients than normal ones. This may be a method of determination of risk factor of LVH patients before doing *** Codes 26A33 Copyright © 2017, The Authors. All rights reserved.

关键词： Cardiology

Application of fractional derivatives in characterization of eCG graphs of right ventricular hypertrophy patients

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arXiv 2017年

There are many functions which are continuous everywhere but non-differentiable at someor all points such functions are termed as unreachable functions. Graphs representing suchunreachable functions are called unreachable graphs. For example eCG is such an unreachable graph. Classical calculus fails in their characterization as derivatives do not exist at the unreachable points. Such unreachable functions can be characterized by fractional calculus as fractional derivatives exist at those unreachable points where classical derivatives do not exist. Definition of fractional derivatives have been proposed by several mathematicians like Grunwald-Letinikov, Riemann-Liouville, Caputo and Jumarie to develop the theory of fractional calculus. in this paper we have used Jumarie type fractional derivative and consequently the phase transition (P.T.) which is the deference between left fractional derivative and right fractional derivatives to characterize those points. A comparative study has been done between normal eCG sample and problematic eCG sample (Right Ventricular Hypertrophy) by the help of above mentioned mathematical *** Codes 26A33 Copyright © 2017, The Authors. All rights reserved.

关键词： Pathology

Revisiting the Curie-Von Schweidler Law for Dielectric Relaxation and Derivation of Distribution Function for Relaxation Rates as Zipf’s Power Law and Manifestation of Fractional Differential equation for Capacitor

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Journal of Modern Physics 2017年第12期8卷 1988-2012页

作者： Shantanu Das Scientist-Reactor Control System Design Section(E&I Group)Bhabha Atomic Research Centre(BARC)MumbaiIndia Honorary Senior Research Professor Condensed Matter Physics Research Centre(CMPRC)Department of PhysicsJadavpur UniversityKolkataIndia

The classical power law relaxation, i.e. relaxation of current with inverse of power of time for a step-voltage excitation to dielectric—as popularly known as Curie-von Schweidler law is empirically derived and is observed in several relaxation experiments on various dielectrics studies since late 19th Century. This relaxation law is also regarded as “universal-law” for dielectric relaxations;and is also termed as power law. This empirical Curie-von Schewidler relaxation law is then used to derive fractional differential equations describing constituent expression for capacitor. in this paper, we give simple mathematical treatment to derive the distribution of relaxation rates of this Curie-von Schweidler law, and show that the relaxation rate follows Zipf’s power law distribution. We also show the method developed here give Zipfian power law distribution for relaxing time constants. Then we will show however mathematically correct this may be, but physical interpretation from the obtained time constants distribution are contradictory to the Zipfian rate relaxation distribution. in this paper, we develop possible explanation that as to why Zipfian distribution of relaxation rates appears for Curie-von Schweidler Law, and relate this law to time variant rate of relaxation. in this paper, we derive appearance of fractional derivative while using Zipfian power law distribution that gives notion of scale dependent relaxation rate function for Curie-von Schweidler relaxation phenomena. This paper gives analytical approach to get insight of a non-Debye relaxation and gives a new treatment to especially much used empirical Curie-von Schweidler (universal) relaxation law.

关键词： Power Law Relaxation Rate Distribution Fractional Derivative Fractional integration Curie-Von Schweidler Law Time-Constants Laplace integral Zipf’s Law integral Representation Time Dependent Relaxation Rate Scale Dependent Relaxation Rate Non-Debye Relaxation

D’Alembert’s solution of fractional wave equations using complex fractional transformation

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arXiv 2017年

作者： Ghosh, Uttam Ali, Md Ramjan Raut, Santanu Sarkar, Susmita Das, Shantanu Department of Applied Mathematics University of Calcutta Kolkata India Mathabhanga College Cooch Behar West Bengal India Reactor Control Systems Design Section E & I Group BARC Mumbai India

Fractional wave equation arises in different type of physical problems such as the vibrating strings, propagation of electro-magnetic waves, and for many other systems. The exact analytical solution of the fractional differential equation is difficult to find. Usually Laplace-Fourier transformation method, along with methods where solutions are represented in series form is used to find the solution of the fractional wave equation. in this paper we describe the D’Alembert’s solution of the fractional wave equation with the help of complex fractional transform method. We demonstrate that using this fractional complex transformation method, we obtain the solutions easily as compared to fractional method of characteristics;and get the solution in analytical form. We show that the solution to the fractional wave equation manifests as travelling waves with scaled coordinates, depending on the considered fractional order value.26A33, 34A08, 35R11 Copyright © 2017, The Authors. All rights reserved.

关键词： Wave equations

investigation of fractional dynamics and design of ultracapacitor based very low frequency oscillators

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Investigation of fractional dynamics and design of ultracapa...

ieee india Conference (iNDiCON)

作者： Mano Ranjan Kumar Subhojit Ghosh Shantanu Das Department of Electrical Engineering National Institute of Technology Raipur Raipur India Reactor Control Systems Design Section E&I Group Bhabha Atomic Research Centre Mumbai India

ISBN: (纸本)9781509036479

in recent years, applications of ultracapacitor in electrical systems have witnessed a rise due to their high power density. Ultracapacitors are known to possess inherent nonlinear dynamics with long term memory, which can be better, described using fractional calculus. This article addresses possible application of ultracapacitors for the design of very low frequency (VLF) oscillators. The presence of fractional dynamics in ultracapacitors has been studied over wide frequency ranges, and utilized to estimate the VLF oscillation frequency. The proposed approach allows design of oscillators for a set of low frequency specifications.

关键词： Supercapacitors Oscillators Mathematical model integrated circuit modeling Optimization Parameter estimation

Fractional Weierstrass Function by Application of Jumarie Fractional Trigonometric Functions and its Analysis

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Advances in Pure Mathematics 2015年第12期5卷 717-732页

作者： Uttam Ghosh Susmita Sarkar Shantanu Das Department of Mathematics Nabadwip Vidyasagar CollegeNabadwipIndia Department of Applied Mathematics University of CalcuttaKolkataIndia Reactor Control Systems Design Section E&I GroupBARCMumbaiIndia

The classical example of no-where differentiable but everywhere continuous function is Weierstrass function. in this paper we have defined fractional order Weierstrass function in terms of Jumarie fractional trigonometric functions. The H?lder exponent and Box dimension of this new function have been evaluated here. it has been established that the values of H?lder exponent and Box dimension of this fractional order Weierstrass function are the same as in the original Weierstrass function. This new development in generalizing the classical Weierstrass function by use of fractional trigonometric function analysis and fractional derivative of fractional Weierstrass function by Jumarie fractional derivative, establishes that roughness indices are invariant to this generalization.

关键词： Holder exponent Fractional Weierstrass Function Box Dimension Jumarie Fractional Derivative Jumarie Fractional Trigonometric Function

design FOR NeW-JeRSeY, iOWA, AND DeS-MOiNeS MODeRNiZATiON

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NAVAL eNGiNeeRS JOURNAL 1984年第3期96卷 25-38页

作者： SiMS, PJ eDWARDS, JR DiCKeY, RL SHULL, HS Philip J. Sims:graduated from Webb Institute in 1971 and went to work for the Advance Design Branch of the Naval Ship Engineering Center. He was part of the FFG-7 design team in 1972. The 1973–75 years were spent developing automated early-stage aircraft carrier design procedures and performing carrier design trade-off work in support of the CVV design. He returned to school in 1976 for a masters at M.I. T. The 1977–80 period was spent updating the Navy's destroyer-cruiser early-stage design procedures and performing studies for the CGN-42 reserve FFX and DDX (later DDG 51) projects. Also during this period he was team leader on concept formulation (CONFORM) studies of new ships such as a heavy combatant and a low detectability ship. From 1981 to early 1983 Mr. Sims was Design Integration Manager for the BB-62 and Ship Design Manager for the BB-61 and CA-134. He is presently principal naval architect for the FFX study and also works on the NA TO frigate effort. James F. Edwards Sr:.is the Technical Director Ship Analytics Inc. Washington D.C. Operations and was the Ship Design Manager for the battleship USSNew Jerseyprior to his departure from NAVSEA in August 1983. He joined the U.S. Navy Reserves in 1954 and served on active duty from 1957 to 1960. From 1961 to 1963 he worked for McLaughlin Research Corporation as a section head in the drafting department. From 1963 to 1966 he worked for the Vitro Corporation of America in the Terrier (surface missile systems) Department. In 1966 he participated in the contract design of the first shipboard integrated digital ASW Command and Control system while working for the Stanwick Corporation. In 1967 Mr. Edwards accepted a position at NAVSHIPS in the Combat System Integration Division. In 1974 he transferred to what is currently NAVSEA's Hull Design Division. In 1980 Mr. Edwards was designated as the Battleship and Heavy Cruiser General Arrangements Task Leader and subsequently served as the Hull Task Group Manager the Ship Configuration Control Manager and fina

in reactivating the battleship New Jersey , the Navy faced three major problems. The baseline data on the ship was not readily available or reliable, a new generation cruise missile armament was proposed, and the ship delivery schedule was very tight. After doing a feasibility study, system design engineers were taken onboard the mothballed ship to resolve the design problems. Being on the ship allowed an intensive effort and immediate reference to the actual ship configuration. The tools used to control this effort were a ship check plan, a ship check form and the master arrangement drawing. Simultaneously with the design effort, a repair scoping effort was conducted. The design evolution and solutions to the major problems are described. The results of the New Jersey effort are shown with sample documentation, the ship characteristics and the downstream design effort. The iowa was the next ship to be modernized. The top level requirements were the same as New Jersey's but new problems were encountered. More options were investigated which diverted attention from the basic effort. A fundamental difference was the iowa had not had a 1968 reactivation as the New Jersey had, so items that were repair and reactivation on the New Jersey in 1968 had to be part of the iowa modernization. A major influence on the iowa design process was that a complete set of specifications for a private yard bid had to be developed. The next effort was to install the same New Jersey modernization payload on a Des Moines class heavy cruiser. Heavy cruisers are large ships but significantly smaller than battleships and much closer to their naval architectural limits of weight and center of gravity. They have much less topside area than the battleships, and the new payload was very topside space consuming. The cruisers are also much more restricted in internal volume. Two feasibility studies were conducted. One resolved volume problems but approached the weight and center of gravity limits.

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iS AUTOMATiON THe MAGiC POTiON FOR MANNiNG PROBLeMS

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NAVAL eNGiNeeRS JOURNAL 1982年第2期94卷 127-136页

作者： MeLLiS, JG PLATO, Ai ReiN, RJ James G. Mellis:attended Central Institute in Kansas City Mo. where he graduated in Electronic Engineering Technology. He later attended the University of Minn. in Minneapolis. At present he works in the Manning and Controls Integration Section of the Naval Sea Systems Command. Mr. Mellis is responsible for developing manpower requirements for ship design and for the coordination of shipboard automation designs with the U.S. Navy's manpower policies and availability. Mr. Mellis is currently developing manpower requirements for the U.S. Navy's DDGX ship design. In this capacity he has examined proposals for shipboard manpower reductions through the use of automation and remote control techniques. Another project where Mr. Mellis is heavily engaged in is the Ship Systems Engineering Standards (SSES) development. Mr. Mellis is the assistant project manager for the test and evaluation and producibility aspects of the SSES project. Previously prior to his employment with NAVSEA Mr. Mellis worked for General Dynamics/Electronic Division as a Senior Field Engineer on the Apollo Instrumentation Ships (i.e. Vanguard Restone Mercury). He was responsible for Central Data Processing Systems on the three ships. Artis I. Plato:is the Head of the Manning and Controls Integration Section of the Naval Sea Systems Command (NAVSEA). He is responsible for the development of accurate manpower requirements for all new construction and major overhaul ship projects for the U.S. NAVY. In addition Mr. Plato must coordinate shipboard controls integration and automation aspects with manpower requirements to insure that a compatible solution is developed. Mr. Plato began his professional career in 1956 at the New York Naval Shipyard. There he worked in the Internal Combustion Engine and Shipboard Elevator Section. During 1957 and 1958 he was called up for active duty with the U.S. Army Corps of Engineers. He served in Europe with various Construction Engineers units. After release from active duty he returned to the shipyard. In 19

This paper examines the recent experience in the UNiTeD STATeS NAVY where automation has been introduced into new ship designs. While other attributes are recognized in the introduction of automated shipboard systems, such as the ability to respond more quickly in combat situations, this paper focuses on the effects of automation upon ship manpower requirements. Specific examples show that expected reductions in manning were not achieved in recent ship designs where automation was incorporated for that purpose. While the use of shipboard automation is not without its critics, the U.S. Fleet appears to have accepted the concept. User feedback addresses the issues of reliability, the provisions for backup systems, the need for better qualified personnel and the concern about maintenance workload. The authors provide specific recommendations for improved guidance to ship designers to more effectively apply automation in the ship design process.

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