SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger's Equations

Abstract The space-time fractional telegraph equation and the space-time fractional modified third-order Kdv equations are significant molding equations in theoretic physics, mathematical physics, plasma physics also other fields of nonlinear sciences. The space time-fractional telegraph equation, which appears in the investigation of an electrical communication line and includes voltage in addition to current which is dependent on distance and time, is also applied to communication lines of wholly frequencies, together with direct current, as well as high-frequency conductors, audio frequency (such as telephone lines), and low frequency (for example cable television) used in the extension of pressure waves into the lessons of pulsatory blood movement among arteries also the one-dimensional haphazard movement of bugs towards an obstacle. The presence of chain rule and the derivative of composite functions allows the nonlinear fractional differential equations (NLFDEs) to translate into the ordinary differential equation employing wave alteration. To explore such categories of resolutions, the extended tanh-method is accomplished via Conformable fractional derivatives. A power sequence in tanh was originally used as an ansatz to provide analytical solutions of the traveling wave type of certain nonlinear evolution equations. To convert this problem to a standard differential equation, a partial complex transformation that has been summarized succinctly is utilized correctly thus, with all of the free parameters, numerous exact logical arrangements are required. The results are found as hyperbolic and rational functions involving parameters, when specific values are supplied to the parameters solitary wave solutions are formed from traveling wave solutions. The outcomes achieved in this study are king type, single soliton, double soliton, multiple solitons, bell shape, and other sorts of forms and we demonstrated that these solutions were validated through the Maple software. The proposed approach for solving nonlinear fractional partial differential equations has been developed to be operative, unpretentious, quick, and reliable to usage.

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An instability of the standard model of cosmology creates the anomalous acceleration without dark energy

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0887 ◽

2017 ◽

Vol 473 (2207) ◽

pp. 20160887

Author(s):

Joel Smoller ◽

Blake Temple ◽

Zeke Vogler

Keyword(s):

Dark Energy ◽

Cosmological Constant ◽

Phase Portrait ◽

Simple Wave ◽

Space Time ◽

Spherically Symmetric ◽

Third Order ◽

Rest Point ◽

Spherically Symmetric Solutions ◽

Anomalous Acceleration

We identify the condition for smoothness at the centre of spherically symmetric solutions of Einstein’s original equations without the cosmological constant or dark energy. We use this to derive a universal phase portrait which describes general, smooth, spherically symmetric solutions near the centre of symmetry when the pressure p =0. In this phase portrait, the critical k =0 Friedmann space–time appears as a saddle rest point which is unstable to spherical perturbations. This raises the question as to whether the Friedmann space–time is observable by redshift versus luminosity measurements looking outwards from any point. The unstable manifold of the saddle rest point corresponding to Friedmann describes the evolution of local uniformly expanding space–times whose accelerations closely mimic the effects of dark energy. A unique simple wave perturbation from the radiation epoch is shown to trigger the instability, match the accelerations of dark energy up to second order and distinguish the theory from dark energy at third order. In this sense, anomalous accelerations are not only consistent with Einstein’s original theory of general relativity, but are a prediction of it without the cosmological constant or dark energy.

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Third-order Killing tensors in the Schwarzschild space-time

General Relativity and Gravitation ◽

10.1007/bf00756193 ◽

1982 ◽

Vol 14 (1) ◽

pp. 27-30 ◽

Cited By ~ 3

Author(s):

Giacomo Caviglia ◽

Clara Zordan

Keyword(s):

Space Time ◽

Third Order ◽

Killing Tensors

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Высокодисперсионные электростатические зеркала вращательной симметрии с времяпролетной фокусировкой по энергии третьего порядка

Журнал технической физики ◽

10.21883/jtf.2020.07.49456.280-19 ◽

2020 ◽

Vol 90 (7) ◽

pp. 1195

Author(s):

С.Б. Бимурзаев ◽

Н.У. Алдияров ◽

З.С. Саутбекова

Keyword(s):

Rotational Symmetry ◽

Time Of Flight ◽

Fundamental Solutions ◽

Space Time ◽

Particle Energy ◽

Third Order ◽

Coaxial Cylinders ◽

Gaussian Plane ◽

Chromatic Aberrations ◽

Axis Of Symmetry

Expressions describing the time of flight of charged particles taking into account time-of-flight chromatic aberrations in electrostatic mirrors of rotational symmetry are presented. The conditions of time-of-flight focusing by particle energy up to the third order inclusively in the three-electrode mirrors, the electrodes of which are coaxial cylinders of equal diameter, were determined by numerical calculations. Various modes of operation of such mirrors are considered: 1) space-time-of-flight focusing in the Gaussian plane of the mirror; 2) space-time-of-flight focusing in the focal plane of the mirror; 3) time-of-flight focusing in telescopic mode; 4) time-of-flight focusing in collimator mode. The results of calculations are presented for two fundamental solutions of the equation of paraxial trajectories. One of these solutions describes trajectories whose direct and reverse branches coincide, and the second solution describes trajectories whose direct and reverse branches are symmetrical with respect to the axis of symmetry of the field. It is shown that the time-of-flight dispersion of the mirror by mass, determined on the basis of the second solution, is several times higher than that obtained on the basis of the first solution for all modes of its operation.

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Probe size and 5th-order aberration

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100125609 ◽

1987 ◽

Vol 45 ◽

pp. 134-135 ◽

Cited By ~ 1

Author(s):

Zhifeng Shao

Keyword(s):

Electron Probe ◽

Spherical Aberration ◽

Wave Length ◽

Cylindrical Symmetry ◽

Electron Wave ◽

Third Order ◽

The Third ◽

Probe Radius ◽

Small Probe ◽

Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

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Space, Time and Einstein

10.1017/upo9781844653447 ◽

2002 ◽

Author(s):

J. B. Kennedy

Keyword(s):

Space Time

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Children’s Recall of Approximations to English

Journal of Speech and Hearing Research ◽

10.1044/jshr.1602.201 ◽

1973 ◽

Vol 16 (2) ◽

pp. 201-212 ◽

Cited By ~ 3

Author(s):

Elizabeth Carrow ◽

Michael Mauldin

Keyword(s):

Language Development ◽

Fourth Order ◽

Third Order ◽

Memory Processes ◽

The Third ◽

General Index ◽

General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

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