Gravitational Field Research Articles

Modified TRIAD Method for Solving the Problem of a Moving Object Orientation

Currently, it is relevant to create moving objects orientation systems based on the integration of various types of primary information sensors. One way to solve the orientation problem is to use the TRIAD (Tri-Axial Orientation Determination) method which allows determining the matrix of direction cosines between two coordinates. The traditional TRIAD force method is based on the use of two reference vectors – of gravitational and geomagnetic fields, measured by accelerometers and magnetometers, respectively. The disadvantage of this method is – appearance of additional deviations during the accelerated movement of the object and influence of primary information sensors’ random errors. Modified TRIAD method which is based on measurements of three triads of sensors: magnetometers, accelerometers and gyroscopes was proposed in the article. Estimates of acceleration vectors of gravitational and geomagnetic fields were calculated taking into account gyroscope measurements. Then these estimates were combined with the accelerometers’ and magnetometers’ data. The complex gravitational and geomagnetic fields’ accelerations were used to form the direction cosine matrix by the TRIAD method. The suggested modified method can be used to implement free-form moving objects’ orientation systems, since it is 6–8 times more accurate compared to the classic TRIAD method. Attenuation of random sensor errors and disturbances due to object acceleration can be adjusted by use of weigh factors.

Correlations and Signaling in the Schrödinger-Newton Model

Abstract The Schrödinger-Newton model is a semi-classical theory in which, in addition to mutual attraction, massive quantum particles interact with their own gravitational fields. While there are many studies on the phenomenology of single particles, correlation dynamics in multipartite systems is largely unexplored. Here, we show that the Schrödinger-Newton interactions preserve the product form of the initial state of a many-body system, yet on average agreeing with classical mechanics of continuous mass distributions. This leads to a simple test of the model, based on verifying bipartite gravitational evolution towards non-product states. We show using standard quantum mechanics that, with currently accessible single-particle parameters, two masses released from harmonic traps get correlated well before any observable entanglement is accumulated. Therefore, the Schrödinger-Newton model can be tested with setups aimed at observation of gravitational entanglement with significantly relaxed requirements on coherence time. We also present a mixed-state extension of the model that avoids superluminal signaling.

Optical aspects of Born-Infeld BTZ black holes in massive gravity

We explore the dynamics of thin accretion disks, the radius of black hole shadows, observed intensities, and the visual characteristics of Born-Infeld BTZ black holes in massive gravity. We find out the relations for angular velocity, specific energy, and angular momentum of particles around the black hole. We observe that intense Born-Infeld electromagnetic effects lead to a reduction in the rotational motion of particles within the accretion disk, and the massive gravity slows down the orbital motion of these particles. We reveal that the influence of massive gravity parameter correlates with a reduction in the black hole’s shadow size, which suggests that massive gravity effects intensify the gravitational fields, thereby reducing the angular diameter of the shadows. On the other hand, a higher Born-Infeld parameter enlarges the black hole’s shadow, which manifests a visual relationship between the black hole’s physical dimensions and its gravitational influence. Moreover, we also uncover the optical characteristics of Born-Infeld BTZ black holes, which show that the Born-Infeld parameter greatly influences the electromagnetic field around the black hole, which affects energy distribution in the space. Finally, we observe that massive gravity significantly influences the spacetime structure near black holes, which is crucial for grasping gravitational lensing and the dynamics of accretion disks under such extreme conditions.

Gravitational waves of nonextremal Kerr black holes from conformal symmetry

In the low-energy limit, the near region of a generic Kerr black hole has been conjectured to be holographically dual to a two-dimensional conformal field theory. In this paper, we consider a test object orbiting in the near region of a nonextremal Kerr black hole. We first couple it to a massless scalar field. The resulting scalar radiation at the horizon is computed from the perspectives of gravity and dual conformal field theory. Considering the influence of nonzero temperature, the agreement of both computations is found. Then, we generalize the analysis to the gravitational radiation and find agreement again. The agreement supports the conjectured holography and provides a potential theoretical tool for gravitational wave computations. Published by the American Physical Society 2024

System Tautochronic Path for Pendulum Vibration Absorber Based on Simple Harmonic Oscillator Transformation

Abstract Centrifugal pendulum vibration absorbers (CPVAs) are essentially collections of pendulums attached to a rotating component or components within a mechanical system for the purpose of mitigating the typical torsional surging that is inherent to internal combustion engines and electric motors. The dynamic stability and performance of CPVAs are highly dependent on the motion path defined for their pendulous masses. Assemblies of absorbers are tuned by adjusting these paths such that the pendulums respond to problematic orders (multiples of average rotation speed) in a way that smooths the rotational accelerations arising from combustion or other nonuniform rotational acceleration events. For most motion paths, pendulum tuning indeed shifts as a function of the pendulum response amplitude. For a given motion path, the tuning shift that occurs as pendulum amplitude varies produces potentially undesirable dynamic instabilities. Large amplitude pendulum motion that mitigates a high percentage of torsional oscillation while avoiding instabilities brought on by tuning shift introduces complexity and hazards into CPVA design processes. Therefore, identifying pendulum paths whose tuning order does not shift as the pendulum amplitude varies, so-called tautochronic paths, may greatly simplify engineering design processes for generating high-performing CPVAs. To illustrate this new approach and results, a tautochronic cutout shape producing constant period system motion is obtained for a simplified problem involving a mass sliding in the cutout of a larger mass that is free to translate horizontally without friction in a constant gravitational field, where the translating base mass replaces the rotating rotor in the centrifugal field.

Effect of Variable Gravity Field on Dual Component Convection in a Couple Stress Fluid Saturated Anisotropic Porous Layer With Temperature‐Dependent Heat Source

ABSTRACTThis study examines how gravity fluctuations, the couple stress parameter, anisotropic parameters, and heat source collectively influence dual‐component convection in the porous layer. The linear analysis is conducted utilizing the normal mode technique. The authors proposed three categories of gravity fluctuation, namely: (a) linear, (b) parabolic, and (c) exponential. Expressions for both stationary and oscillatory Rayleigh numbers are derived using the Galerkin approach. Neutral stability curves for both stationary and oscillatory modes are analyzed, with graphical representations to show the effects of various stability parameters, including gravity fluctuations, couple stress, anisotropy, and heat source. The results show that the mechanical anisotropy parameter and Vadasz number lead to system destabilization, while the couple stress parameter, Lewis number, gravity parameter, solute Rayleigh number, and thermal anisotropy parameter help to stabilize the system. Furthermore, the system is more stable with exponential gravity fluctuations and less stable with parabolic gravity fluctuations. This finding offers insights into thermal convective instability in porous media, impacting applications in geoscience, engineering, and environmental science.

Weighted Fusion Method of Marine Gravity Field Model Based on Water Depth Segmentation

Among the marine gravity field models derived from satellite altimetry, the Scripps Institution of Oceanography (SIO) series and Denmark Technical University (DTU) series models are the most representative and are often used to integrate global gravity field models, which were inverted by the deflection of vertical method and sea surface height method, respectively. The fusion method based on the offshore distance used in the EGM2008 model is just model stitching, which cannot realize the true fusion of the two types of marine gravity field models. In the paper, a new fusion method based on water depth segmentation is proposed, which established the Precision–Depth relationship of each model in each water depth segment in the investigated area, then constructed the FUSION model by weighted fusion based on the precision predicted from the Precision–Depth relationship at each grid in the whole region. The application in the South China Sea shows that the FUSION model built by the new fusion method has better accuracy than SIO28 and DTU17, especially in shallow water and offshore areas. Within 20 km offshore, the RMS of the FUSION model is 5.10 mGal, which is 8% and 4% better than original models, respectively. Within 100 m of shallow water, the accuracy of the FUSION model is 4.01 mGal, which is 14% and 12% higher than the original models, respectively. A further analysis shows that the fusion model is in better agreement with the seabed topography than original models. The new fusion method can blend the effective information of original models to provide a higher-precision marine gravity field.

Precise Geoid Determination in the Eastern Swiss Alps Using Geodetic Astronomy and GNSS/Leveling Methods

Astrogeodetic deflections of the vertical (DoVs) are close indicators of the slope of the geoid. Thus, DoVs observed along horizontal profiles may be integrated to create geoid undulation profiles. In this study, we collected DoV data in the Eastern Swiss Alps using a Swiss Digital Zenith Camera, the COmpact DIgital Astrometric Camera (CODIAC), and two total station-based QDaedalus systems. In the mountainous terrain of the Eastern Swiss Alps, the geoid profile was established at 15 benchmarks over a two-week period in June 2021. The elevation along the profile ranges from 1185 to 1800 m, with benchmark spacing ranging from 0.55 km to 2.10 km. The DoV, gravity, GNSS, and levelling measurements were conducted on these 15 benchmarks. The collected gravity data were primarily used for corrections of the DoV-based geoid profiles, accounting for variations in station height and the geoid-quasigeoid separation. The GNSS/levelling and DoV data were both used to compute geoid heights. These geoid heights are compared with the Swiss Geoid Model 2004 (CHGeo2004) and two global gravity field models (EGM2008 and XGM2019e). Our study demonstrates that absolute geoid heights derived from GNSS/leveling data achieve centimeter-level accuracy, underscoring the precision of this method. Comparisons with CHGeo2004 predictions reveal a strong correlation, closely aligning with both GNSS/leveling and DoV-derived results. Additionally, the differential geoid height analysis highlights localized variations in the geoid surface, further validating the robustness of CHGeo2004 in capturing fine-scale geoid heights. These findings confirm the reliability of both absolute and differential geoid height calculations for precise geoid modeling in complex mountainous terrains.

Influence of GUP corrected Casimir energy on zero tidal force wormholes in modified teleparallel gravity with matter coupling

In recent times, the study of the Casimir effect in quantum field theory has garnered increasing attention because of its potential to be an ideal source of exotic matter needed for stabilizing traversable wormholes. It has been confirmed through experimental evidence that this phenomenon involves fluctuations in the vacuum field, leading to a negative energy density. Motivated by the above, we have investigated Casimir wormholes with corrections from the Generalized Uncertainty Principle (GUP) within the framework of matter-coupled teleparallel gravity. Our analysis includes three well-known GUP models: the Kempf, Mangano, and Mann (KMM) model, the Detournay, Gabriel, and Spindel (DGS) model, and a third model called Model II. For a broader analysis, we have considered two well-known model functions for the teleparallel theory: a linear f(T,T)=αT+βT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(T,\\mathcal {T})=\\alpha T+\\beta \\mathcal {T}$$\\end{document} and a quadratic model f(T,T)=ηT2+χT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(T,\\mathcal {T})=\\eta T^2+\\chi \\mathcal {T}$$\\end{document}. The shape function solutions corresponding to both models are examined in the absence of tidal forces in spacetime. We also demonstrate the crucial role played by the parameters of the f(T,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(T,\\mathcal {T})$$\\end{document} models in the violation of the energy conditions. With the increasing interest in detecting gravitational waves from astrophysical objects, we have thoroughly discussed the perturbation of the wormhole solutions in the scalar, electromagnetic, axial gravitational, and Dirac field backgrounds. We employ the 3rd order WKB expansion to find the complex frequencies associated with the quasinormal modes of energy dissipation. Additionally, we also calculate the active mass and total gravitational energy for the wormhole geometry. The amount of exotic matter involved in sustaining these wormholes is also found in this paper. Furthermore, the physical stability of such Casimir wormholes is examined using the Tolman–Oppenheimer–Volkoff equation.

McVittie–Plummer Spacetime: Plummer Sphere Immersed in the FLRW Universe

The McVittie–Plummer spacetime is a spherically symmetric inhomogeneous cosmological model that represents a spherical star system embedded in a standard Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological model. We study the main physical properties of this gravitational field. Regarding the interplay between the physics of the local system and the expanding background, we employ the Misner–Sharp mass–energy function to show that there is a relatively weak time-dependent general relativistic coupling between the astrophysical system and the background FLRW cosmological model. The coupling term is proportional to the inverse of the scale factor and decreases as the Universe expands.

Classical characters of spinor fields in torsion gravity

Abstract We consider the problem of having relativistic quantum mechanics re-formulated with hydrodynamic variables, and specifically the problem of deriving the Mathisson-Papapetrou-Dixon equations (describing the motion of a massive spinning body moving in a gravitational field) from the Dirac equation. The problem will be answered on a general manifold with torsion and gravity. We will demonstrate that when plane waves are considered the MPD equations describe the general relativistic wave-particle duality with torsion, but we will also see that in such a form the MPD equations become trivial.

Testing Whether Gravity Acts as a Quantum Entity When Measured

A defining signature of classical systems is “in principle measurability” without disturbance: a feature manifestly violated by quantum systems. We describe a multi-interferometer experimental setup that can, in principle, reveal the nonclassicality of a spatial superposition-sourced gravitational field if an irreducible disturbance is caused by a measurement of gravity. While one interferometer sources the field, the others are used to measure the gravitational field created by the superposition. This requires neither any specific form of nonclassical gravity, nor the generation of entanglement between any relevant degrees of freedom at any stage, thus distinguishing it from the experiments proposed so far. This test, when added to the recent entanglement-witness based proposals, enlarges the domain of quantum postulates being tested for gravity. Moreover, the proposed test yields a signature of quantum measurement induced disturbance for any finite rate of decoherence, and is device independent. Published by the American Physical Society 2024

Searching for Dark Matter Based on Gravitational Lensing

Gravitational lensing is a powerful astronomical tool in which scholars can accurately predict the distribution of dark matter in massive objects by observing the bending effects of light in the gravitational fields of huge objects (e.g., galaxies or galaxy clusters). Since dark matter does not emit directly, or is strongly interacts with ordinary matter, gravitational lensing gives us a unique view of observing and measuring dark matter. This study introduces some principles of gravitational lensing effect to provide favorable evidence in the study of dark matter and other celestial bodies (including an engineering study in the focal region of solar gravitational lensing (SGL) and other experiments). This research will see how the properties of the clear and focused gravitational lens are applied to celestial observations, and how they are captured by the detector. Strong gravitational lensing can better show distant galaxies (including high redshift) around the dark matter distribution, and through the several basic quantities in gravitational lens analytical and numerical calculation and analysis, to simulate the center of the supermassive black hole, and gravitational lensing image and model of dark matter mass can be detected and dark matter has the key evidences of the interaction. It reveals the intrinsic and distribution of dark matter, thus driving the understanding of the universe’s evolution and its fundamental components. Therefore, gravitational lensing has an irreplaceable value and a far-reaching influence in exploring the unsolved mystery of dark matter.

Analysis of the Forming for Black Hole and Connections with Timespace

Black holes remain the most popular topic in astronomy to this day because of the mystery that makes it so thoughtprovoking. It forms when a star collapses or a huge mass gather. The greater the mass of the black hole, the greater the gravitational field and the faster the galaxies contract, and eventually a massive galaxy will be swallowed up by this black hole and form a singularity. Their immense mass can bend and distort the space-time around them, and this distorting effect leads to a gravitational trap that makes it impossible for anything approaching the black hole to escape its gravitational pull, not even light or radiation. This study begins with an introduction to the basic formation principles and structure of black holes and introduces why it is related to space-time, followed by a description of how black holes are observed, their limitations, and future prospects. According to the analysis, one can acquire more knowledge about this mysterious object and lay the foundation for future research on black holes.

Mathematical model for highly sensitive photonic crystal fiber sensor based on hyperbolic black holes

In this paper, the photonic crystal fiber sensor based on the geometry of hyperbolic black holes is proposed. As the hyperbolic black hole concentrates the electromagnetic radiation in its powerful gravitational field, the designed sensor has resulted in the concentration of the most electromagnetic field in the core of fiber. The extraordinary sensitivity of the sensor is due to the topology and exact geometry, which originates from the idea of the black hole. All the parameters related to the geometry of the structure are optimized by the Nelder-Mead algorithm, also, a unique three variable equation for the behavior of the structure is provided, which allows obtaining the possible errors due to the construction based on the geometry of the structure. In addition, the calculated equation leads to saving the simulation time. The proposed sensor has an amplitude sensitivity of 4650 (RIU-1) and wavelength sensitivity of 7000 (nm/RIU) and covers a wide range of refractive indices (1.31–1.42) in the bio and chemical range. Because of the amazing options that represent the functionality of the sensor, its construction is highly recommended.

On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC

Abstract In spatially non-compact homogeneous minisuperpace models, spatial integrals in the Hamiltonian and symplectic form must be regularised by confining them to a finite volume $V_o$, known as the \emph{fiducial cell}. As this restriction is unnecessary in the complete field theory before homogeneous reduction, the physical significance of the fiducial cell has been largely debated, especially in the context of (loop) quantum cosmology. Understanding the role of $V_o$ is in turn essential for assessing the minisuperspace description's validity and its connection to the full theory. In this work we present a systematic procedure for the field theory reduction to spatially homogeneous minisuperspaces within the canonical framework and apply it to a massive scalar field theory and gravity. Our strategy consists in implementing spatial homogeneity via second-class constraints for the discrete field modes over a partitioning of the spatial slice into countably many disjoint cells. The reduced theory's canonical structure is then given by the corresponding Dirac bracket. Importantly, the latter can only be defined on a finite number of cells homogeneously patched together. This identifies a finite region, the fiducial cell, whose physical size acquires then a precise meaning already at the classical level as the scale over which homogenenity is imposed. Additionally, the procedure allows us to track the information lost during homogeneous reduction and how the error depends on $V_o$. We then move to the quantisation of the classically reduced theories, focusing in particular on the relation between the theories for different $V_o$, and study the implications for statistical moments, quantum fluctuations, and semiclassical states. In the case of a quantum scalar field, a subsector of the full quantum field theory where the results from the ``first reduced, then quantised" approach can be reproduced is identified and the conditions for this to be a good approximation are determined.

QPOs from charged particles around magnetized black holes in braneworlds

Quasiperiodic oscillations (QPOs) are a powerful tool for testing gravity theories, probing gravitational and electromagnetic field properties, and obtaining constraints on the black hole and field parameters. This work considers charged particle dynamics near uniformly magnetized black holes in braneworlds. First, we obtain the solution of the Maxwell equation for magnetic fields and calculate the radial and angular magnetic field components. We derive and analyze the effective potential of charged particles for circular orbits and investigate the energy and angular momentum for the circular orbits. We also analyze the combined effects of magnetic interaction and braneworlds on the charged particles’ innermost stable circular orbits (ISCOs). We calculate the angular momentum of charged particles in Keplerian orbits in the presence of an external magnetic field and braneworlds. Also, we investigate frequencies of the particle oscillations along vertical and angular directions. We applied our studies on particle oscillations to the QPO studies in the relativistic precession model. Finally, we obtain constraints on magnetic interaction and braneworld parameters together with the black hole mass and QPO orbits using Monte Carlo Markov Chain (MCMC) simulation in the four-dimensional parameter space for the QPOs observed in the microquasars XTE J1550-564, GRO J1655-40 & GRS 1915-105, and at the center of galaxies M82 and Milky Way.

An X-ray high-frequency quasi-periodic oscillation in NGC 1365

This study presents the detection of a high-frequency quasi-periodic oscillation (QPO) in the Seyfert galaxy NGC 1365 based on observational data obtained by XMM-Newton in January 2004. Utilizing the weighted wavelet Z-transform (WWZ) and Lomb-Scargle periodogram (LSP) methods, a QPO signal is identified at a frequency of 2.19 × 10−4 Hz (4566 s), with a confidence level of 3.6σ. The signal is notably absent in the lower 0.2–1.0 keV energy band, with the primary contribution emerging from the 2.0–10.0 keV band, where the confidence level reaches 3.9σ. Spectral analysis shows that there are multiple absorption and emission lines in the high-energy band (> 6 keV). The correlation between the QPO frequency (fQPO) and the mass of the central black hole (MBH) of NGC 1365 aligns with the established logarithmic trend observed across black holes, indicating the QPO is of high frequency. This discovery provides new clues for studying the generation mechanism of QPOs in Seyfert galaxies, which helps us understand the accretion process around supermassive black holes and the characteristics of strong gravitational fields in active galactic nuclei.

Impacts of Digital Elevation Model Elevation Error on Terrain Gravity Field Calculations: A Case Study in the Wudalianchi Airborne Gravity Gradiometer Test Site, China

Accurate Digital Elevation Models (DEMs) are essential for precise terrain gravity field calculations, which are critical in gravity field modeling, airborne gravimeter and gradiometer calibration, and geophysical inversion. This study evaluates the accuracy of various satellite DEMs by comparing them with a LiDAR DEM at the Wudalianchi test site, a location requiring ultra-accurate terrain gravity fields. Major DEM error sources, particularly those related to vegetation, were identified and corrected using a least squares method that integrates canopy height, vegetation cover, NDVI, and airborne LiDAR DEM data. The impact of DEM vegetation errors on terrain gravity anomalies and gravity gradients was quantified using a partitioned adaptive gravity forward-modeling method at different measurement heights. The results indicate that the TanDEM-X DEM and AW3D30 DEM exhibit the highest vertical accuracy among the satellite DEMs evaluated in the Wudalianchi area. Vegetation significantly affects DEM accuracy, with vegetation-related errors causing an impact of approximately 0.17 mGal (RMS) on surface gravity anomalies. This effect is more pronounced in densely vegetated and volcanic regions. At 100 m above the surface and at an altitude of 1 km, vegetation height affects gravity anomalies by approximately 0.12 mGal and 0.07 mGal, respectively. Additionally, vegetation height impacts the vertical gravity gradient at 100 m above the surface by approximately 4.20 E (RMS), with errors up to 48.84 E over vegetation covered areas. The findings underscore the critical importance of using DEMs with vegetation errors removed for high-precision terrain gravity and gravity gradient modeling, particularly in applications such as airborne gravimeter and gradiometer calibration.

Shock wave propagation in a real gas with or without gravitational field in the presence of magnetic field and monochromatic radiation via group invariance method

PurposeThis article aims to find the similarity solutions for the one-dimensional motion of spherical symmetric shock wave in non-ideal gas influenced by the azimuthal magnetic field and monochromatic radiation in the presence or absence of gravitational field. This paper also aims to study the effects of physical parameters on the strength of shock wave, and on the flow variables in the flow-field region behind the shock front.Design/methodology/approachThe Roche model is used to describe the gravitational field effects due to a massive nucleus at the point of symmetry. To derive the similarity solutions, the Lie group symmetry method has been used. Also, the numerical solutions to the present problem are obtained by using Rung–Kutta method of the fourth order with the use of Mathematica software. The effects of variation in the parameter of non-idealness of the gas, the gravitation parameter, the strength of the ambient magnetic field and the adiabatic index of the gas on the shock wave, and on the flow variables is discussed. A comparative study between with and without gravitational field is also, made.FindingsFor different choices of the arbitrary constants that appeared in the solution of infinitesimal generators, we have obtained seven distinct cases of similarity solutions. In the absence of the gravitational field, the similarity solution exists to the power and exponential law shock paths, but in the presence of gravitational field, the similarity solution exists to the power law shock path case only. In the absence of gravitational field, the shock strength is enhanced in the exponential law shock path case in comparison to the power law shock path case. It is found that the shock wave decays with an increase in the value of the adiabatic exponent, the strength of magnetic field, non-idealness of the gas or gravitational parameter.Research limitations/implicationsThe consideration of medium under the influence of gravitational field due to a heavy nucleus at the center and presence of magnetic field decrease the shock strength. This result may be helpful in designing space vehicle and jet engine.Practical implicationsThe result of the present study may be used in the analysis of data from the measurements by space craft in the solar wind and in neighborhood of the Earth’s magnetosphere.Social implicationsThe obtained results may be used for mankind.Originality/valueThe study of spherical shock wave propagation influenced by monochromatic radiation and azimuthal magnetic field in a non-ideal gas with or without gravitational field has yet to be discussed by any authors by using the Lie group symmetry method. In this article, we have discussed all possible cases of similarity solutions using the Lie group symmetry method, which is not studied by anyone as known to us.