Free Parameters Research Articles

The PDI model system for parameterizing soil hydraulic properties

AbstractThe Peters–Durner–Iden (PDI) model system for describing soil hydraulic properties (SHP) has been developed over a decade. Inspired by Rien van Genuchten's seminal work, the PDI system focuses on an efficient and simple parameterization of water retention curves and hydraulic conductivity curves (HCC) across the entire soil moisture spectrum. By combining capillary and non‐capillary components for water retention and conductivity, it aims to reconcile mathematical simplicity and insights on water adsorption and film flow in soils. Recent developments have reduced the number of free parameters of the conductivity model to zero, enhancing the model's applicability in cases of limited data availability. The first reduction was achieved by a prediction of absolute non‐capillary conductivity based on the consideration of film and corner flow on the pore scale, and the second by a prediction of absolute capillary conductivity by a capillary bundle model. This allows a complete characterization of SHP over the entire moisture range with only four retention curve parameters. The inclusion of a maximum pore size in the capillary conductivity model prevents an unrealistic drop of the HCC near saturation. This paper provides a comprehensive overview of the PDI model system, emphasizing its conceptual features and mathematical details. An Excel sheet and a Python code stored in a repository are provided for accessibility.

Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

Determination of gas discharge plasma potential by Langmuir probe using different methods

In a hollow cathode discharge in helium at reduced pressure, the profile of the plasma potential Up in the cathode-anode region was measured with a single Langmuir probe. A digital technique was used to control and analyze the experimental results. The probe current-voltage characteristic was generated by applying a voltage with a spectrum with an equal probability distribution in the measured voltage range. Three methods were compared: (1) differentiation of the probe current-voltage characteristic (IVC) by the Langmuir-Druyvesteyn (LD) method with the Savitsky-Golay (SG) filter; (2) definition of the position of the noise maximum in the experimental IVC; (3) determination of the voltage corresponding to the intersection of the asymptotes of different range of the IVC. Methods (2) and (3) were used both as alternatives to method (1) for measuring the potential Up and as additional ones when choosing free parameters of the SG filter for plasma diagnostics by method (1).

Entropy-calibrated stellar modeling: Testing and improving the use of prescriptions for the entropy of adiabatic convection

Modeling the convection process is a long-standing problem in stellar physics. To date, all ad hoc models have relied on a free parameter, $ (among others) that has no real physical justification and is therefore poorly constrained. However, a link exists between this free parameter and the entropy of the stellar adiabat. There are existing prescriptions, derived from 3D stellar atmospheric models, that treat entropy as a function of stellar atmospheric parameters (effective temperature, surface gravity, and chemical composition). This can offer sufficient constraints on alpha through the development of entropy-calibrated models. However, several questions have arisen as these models are increasingly used with respect to which prescription should be used and whether it ought to be used in its original form, along with the impacts of uncertainties on entropy-calibrated models. We aim to study the three existing prescriptions in detail and determine which of them demonstrate the most optimal performance and how it should be applied. We implemented the entropy-calibration method into the stellar evolution code ( and performed comparisons with the Sun and the system. In addition, we used data from the CIFIST grid of 3D atmosphere models to evaluate the accuracy of the prescriptions. Of the three entropy prescriptions currently available, we determined the one that has the best functional form for reproducing the entropies of the 3D models. However, the coefficients involved in this formulation must not be taken from the original paper because they were calibrated against a flawed set of entropies. We also demonstrate that the entropy obtained from this prescription should be corrected for the evolving chemical composition and for an entropy offset different between various EoS tables. This must be done following a precise procedure to ensure that the classical parameters obtained from the models are not strongly biased. Finally, we provide a data table with entropy of the adiabat of the CIFIST grid, along with the fits for these entropies. Thanks to a precise examination of entropy-calibrated modeling, we are able to offer our recommendations with respect to which adiabatic entropy prescription to use, how to correct it, and how to implement the method into a stellar evolution code.

Spontaneous persistent activity and inactivity in vivo reveals differential cortico-entorhinal functional connectivity

Understanding the functional connectivity between brain regions and its emergent dynamics is a central challenge. Here we present a theory-experiment hybrid approach involving iteration between a minimal computational model and in vivo electrophysiological measurements. Our model not only predicted spontaneous persistent activity (SPA) during Up-Down-State oscillations, but also inactivity (SPI), which has never been reported. These were confirmed in vivo in the membrane potential of neurons, especially from layer 3 of the medial and lateral entorhinal cortices. The data was then used to constrain two free parameters, yielding a unique, experimentally determined model for each neuron. Analytic and computational analysis of the model generated a dozen quantitative predictions about network dynamics, which were all confirmed in vivo to high accuracy. Our technique predicted functional connectivity; e. g. the recurrent excitation is stronger in the medial than lateral entorhinal cortex. This too was confirmed with connectomics data. This technique uncovers how differential cortico-entorhinal dialogue generates SPA and SPI, which could form an energetically efficient working-memory substrate and influence the consolidation of memories during sleep. More broadly, our procedure can reveal the functional connectivity of large networks and a theory of their emergent dynamics.

Non-Local Cosmology: From Theory to Observations

We examine the key aspects of gravitational theories that incorporate non-local terms, particularly in the context of cosmology and spherical symmetry. We thus explore various extensions of General Relativity, including non-local effects in the action through the function F(R,□−1R), where R denotes the Ricci curvature scalar and the operator □−1 introduces non-locality. By selecting the functional forms using Noether Symmetries, we identify exact solutions within a cosmological framework. We can thus reduce the dynamics of these chosen models and obtain analytical solutions for the equations of motion. Therefore, we study the capability of the selected models in matching cosmological observations by evaluating the phase space and the fixed points; this allows one to further constrain the non-local model selected by symmetry considerations. Furthermore, we also investigate gravitational non-local effects on astrophysical scales. In this context, we seek symmetries within the framework of f(R,□−1R) gravity and place constraints on the free parameters. Specifically, we analyze the impact of non-locality on the orbits of the S2 star orbiting SgrA*.

New exact optical soliton solutions of the derivative nonlinear Schrödinger equation family

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton.

Condenser operating conditions effects on dielectric liquid boiling and condensation heat transfer in closed systems

Localized boiling and enclosed condensation of a dielectric liquid in a thermosyphon can be used as an alternative to traditional air cooling for high performance microprocessors. A small quantity of dielectric fluid boils at the surface of the microprocessor and is condensed by a heat exchanger, which removes the heat from the system. Here, the heat exchanger was a water-fed cold plate cooling a round fins heat sink. Such a thermosyphon can remove large heat fluxes and increase the energy efficiency of data centers.An experimental setup was built to measure the impact of various operating parameters on the thermal resistances of the thermosyphon. The thermal resistances that were considered were the boiling resistance between the heater and the dielectric fluid, the condensation resistance between the dielectric fluid and the cold water feeding the cold plate, and their sum (the total thermal resistance). The operating parameters tested were the cold plate inlet flow rate (0.2 L/min and 1 L/min), the cold plate inlet temperature (15 °C and 24 °C) and the filling ratio of the enclosure (33 %, 66 % and 88 %).It was observed that changing the filling ratio did not have a significant impact on the thermal resistances of the thermosyphon. Increasing the cold plate inlet flow rate from 0.2 L/min to 1 L/min and decreasing the cold plate inlet temperature from 24 °C to 15 °C both led to significantly lower total thermal resistance. However, increasing the flow rate lowered the rate at which the pressure increased with the amount of heat removed from the system, while this rate did not change when decreasing the cold plate inlet temperature.These observations could be properly explained by a simple analytical model, with a single free parameter (increasing the thermal resistance of the heat sink for the lowest cold plate flow rates). With the free parameter adjusted to the data, the model had a root-mean square error of 1.8 °C and a maximum error of 6 °C, over all measured conditions.This study demonstrates how changing the filling ratio or the cold plate inlet temperature and flow rate have an impact on the pressure, the temperature and the thermal resistances inside the thermosyphon. This is a first step in evaluating the energetic and economic efficiency of the thermosyphon solution for its future integration in data centers.

Analytic solutions of Eliashberg gap equations at superconducting critical temperature

We perform simple mathematical analysis of the Eliashberg gap equations, and derive the analytic expression at the superconducting critical temperature. In order to perform exact summation of LTc=∑λm12m−1, we use the Einstein and Debye models for the Eliashberg spectral function. This quantity is in good agreement with data derived from experimental and DFT data. To the leading-term approximation, the analytic expression of the superconducting critical temperature is of the form Tc=2eγ−1πωxexp−1+μ*ln4eγmcλ, where ωx=ω2 or ωln. The frequencies ω2 and ωln come out naturally from the models. These equations contain mc as an adjustable parameter. There are no other free parameters and no empirical formula involved. We compare the results of these equations with the celebrated Allen-Dynes modified McMillan equation together with the data from experiments and DFT calculations. The data show strong correlation between our analytic expression and the celebrated Allen-Dynes equation.

Inverse problem for determining free parameters of a reduced turbulent transport model for tokamak plasma

Inverse problem for determining free parameters of a reduced turbulent transport model for tokamak plasma

Friction Velocity Determination Techniques in Turbulent Boundary Layers with Miniature Vortex Generators

Miniature vortex generators (MVGs) have the potential to control turbulent boundary layers (TBLs). Analyzing the scaled turbulent structure of MVG TBLs in experiments typically requires indirect methods for accurate friction velocity Uτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_\ au$$\\end{document} determination. The established methods for Uτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_\ au$$\\end{document} determination need to be verified for non-generic TBLs, such as MVG TBLs, especially at near-wake stations. This study compares the performance of the five most common Uτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_\ au$$\\end{document}-determination methods when applied to MVG TBLs. The methods are tuned in terms of their free parameters and fitting ranges using direct measurements of Uτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_\ au$$\\end{document} from large eddy simulation data, and applied to a corresponding experiment to assess their accuracy. Based on the findings, an “inner” method, which is the Musker function with a drifting buffer layer incorporating a bump function, is recommended for MVG TBLs of the form investigated, as it provides a good estimation of Uτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_\ au$$\\end{document} with uncertainty <3%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$<3\\%$$\\end{document} for all streamwise stations. The method is applied to three further experimental tests with different flow conditions to study trends in Uτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_\ au$$\\end{document}. The error of the drag variation is <6%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$<6\\%$$\\end{document}, which indicates the method is reliable for MVG TBLs at high Reynolds numbers.

High-order accurate transient and free-vibration analysis of plates and shells

The limited availability of analytical solutions and the high cost associated with experimental testing motivate the use of computational tools to assess the dynamic behavior of load-bearing components, especially when a wide design space must be explored, as is often the case with composite structures. In this context, a novel high-order accurate discontinuous Galerkin formulation for transient and free-vibration analysis of multilayered plates and shells is presented and numerical validated. The starting point of the formulation is a generalized structural theory for multilayered shells with arbitrary curvature based on the expansion of the displacement covariant components throughout the shell thickness. The variational statement of three-dimensional elastodynamics allows deriving the strong form of the governing differential equations, which form the basis to obtain the corresponding discontinuous Galerkin weak statements. As the order of the through-the-thickness expansion and the order of the discontinuous Galerkin basis functions are free parameters, the proposed approach allows tuning the order of accuracy of the computed solution throughout both the shell thickness and the shell modeling domain. Numerical results are reported and discussed for several validation test cases in terms of h- and p-convergence analyses, demonstrating the high-order accuracy, robustness, and computational savings of the formulation.

Attractors in [formula omitted]-gravity

We investigate the asymptotic behavior of the cosmological field equations in Symmetric Teleparallel General Relativity, where a nonlinear function of the boundary term is introduced instead of the cosmological constant to describe the acceleration phase of the universe. Our analysis reveals constraints on the free parameters necessary for the existence of an attractor that accurately represents acceleration. However, we also identify asymptotic solutions depicting Big Rip and Big Crunch singularities. To avoid these solutions, we must impose constraints on the phase-space, requiring specific initial conditions.

Predicting the Slowing of Stellar Differential Rotation by Instability-driven Turbulence

Differentially rotating stars and planets transport angular momentum (AM) internally due to turbulence at rates that have long been a challenge to predict reliably. We develop a self-consistent saturation theory, using a statistical closure approximation, for hydrodynamic turbulence driven by the axisymmetric Goldreich–Schubert–Fricke instability at the stellar equator with radial differential rotation. This instability arises when fast thermal diffusion eliminates the stabilizing effects of buoyancy forces in a system where a stabilizing entropy gradient dominates over the destabilizing AM gradient. Our turbulence closure invokes a dominant three-wave coupling between pairs of linearly unstable eigenmodes and a near-zero frequency, viscously damped eigenmode that features latitudinal jets. We derive turbulent transport rates of momentum and heat and provide them in analytic forms. Such formulae, free of tunable model parameters, are tested against direct numerical simulations; the comparison shows good agreement. They improve upon prior quasi-linear or “parasitic saturation” models containing a free parameter. Given model correspondences, we also extend this theory to heat and compositional transport for axisymmetric thermohaline-instability-driven turbulence in certain regimes.

Constraints on bulk viscosity in [formula omitted] gravity from H(z)/Pantheon+ data

In this study, we investigate the role of bulk viscosity in f(Q,T) gravity in explaining late-time cosmic acceleration. This model, an extension of symmetric teleparallel gravity, introduces viscosity into cosmic matter dynamics for a more realistic representation. Specifically, we consider the linear form of f(Q,T)=αQ+βT, where α and β are free model parameters. To assess the model, we derive its exact solution and use Hubble parameter H(z) data and Pantheon + SNe Ia data for parameter estimation. We employ the χ2 minimization technique alongside the MCMC random sampling method to determine the best-fit parameters. Then, we analyze the behavior of key cosmological parameters, including the deceleration parameter, bulk viscous matter-dominated universe density, effective pressure, and the effective EoS parameter, accounting for the viscous type fluid. We observe a transition in the deceleration parameter from a positive (decelerating) to a negative (accelerating) phase at transition redshift zt. The matter density shows the expected positive behavior, while the pressure, influenced by viscosity, exhibits negative behavior, indicative of accelerating expansion. Furthermore, we investigate the energy conditions and find that while the NEC and DEC meet positivity criteria, the SEC is violated in the present and future epochs. The Om(z) diagnostic suggests that our model aligns with quintessence behavior. Finally, our f(Q,T) cosmological model, incorporating bulk viscosity effects, provides a compelling explanation for late-time cosmic behavior, consistent with observational data.

On the existence and stability of traversable wormhole solutions with novel shapefunctions in the framework of F(R,Lm) gravity

In this work, we have explored wormhole (WH) solutions in F(R,Lm) gravity by assuming the Morris–Thorne WH metric and F(R,Lm)=R2+(1+γR)Lm , where γ is the free model parameter. We determined the WH solutions by utilizing two newly developed shape functions (SF) that satisfy all basic conditions for a WH’s physical validity. We also observe that the null energy condition (NEC) behaves negatively. Finally, for both models, we use the volume integral quantifier ( VIQ ) and Tolman–Oppenheimer–Volkoff (TOV) equation to determine how much exotic matter is needed near the WH throat and the stability of the WH. The extensive detailed discussions of the matter components have been done via graphical analysis. The obtained WH geometries meet the physically acceptable conditions for a stable wormhole.

Satellite-derived equilibrium shoreline modelling at a high-energy meso-macrotidal beach

Modelling and predicting the future of sandy shorelines is a key challenge in coastal research and is critical for sustainable coastal management. However, currently the most skillful shoreline models strongly rely on data to calibrate the free parameters, and are thus restricted to a few well monitored sites in the world. Here we address the challenges and opportunities offered by optical satellite imagery to provide useful information for equilibrium shoreline model calibration on cross-shore transport dominated sites. We focus on Truc Vert beach, southwest France, where previous work showed good equilibrium model skill to reproduce shoreline change from the time scales of hours (storms) to decades. Satellite derived waterlines are extracted over 11 years (2009–2020) and further transformed into satellite derived shorelines (SDS) with different water level corrections (e.g. tide and/or run up) and varying alongshore averaging lengths, and thus different uncertainties, in order to test model performance. Successively the timeseries duration and sampling frequency required for model calibration were also investigated. The model calibrated using the SDS data showed similar skill as the model calibrated using in-situ alongshore averaged shoreline positions, even for the uncorrected SDS dataset which Root Mean Square Error (RMSE) are approximately 30 m. Alongshore averaging was found to be the only necessary processing of the SDS data while any other site-specific corrections did not significantly improve model skill. Finally to further investigate the effect of sampling frequency and noise in the dataset we performed an analysis using a synthetic shoreline. Our results suggest that the effect of noise is negligible as long as the sampling frequency remains high (dt ≤ 30 days). Pending further validation, results show the strong potential of using uncorrected SDS dataset for shoreline model calibration at cross-shore transport dominated sandy coasts.

EMPRESS. XIV. Strong High-ionization Lines of Young Galaxies at z = 0–8: Ionizing Spectra Consistent with the Intermediate-mass Black Holes with M BH ∼ 103–106 M ⊙

We present ionizing spectra estimated at 13.6–100 eV for 10 dwarf galaxies with strong high-ionization lines of He ii λ4686 and [Ne v] λ3426 ([Ne iv] λ2424) at z = 0 (z = 8) that are identified in our Keck/LRIS spectroscopy and the literature (the JWST Early Release Observations program). With the flux ratios of these high-ionization lines and >10 low-ionization lines of hydrogen, helium, oxygen, neon, and sulfur, we determine ionizing spectra consisting of stellar and nonthermal power-law radiation by photoionization modeling with free parameters of nebular properties, including metallicity and ionization parameter, canceling out abundance ratio differences. We find that all of the observed flux ratios are well reproduced by the photoionization models with the power-law index α EUV of α EUV ∼ (–1) − 0 and the luminosity L EUV of L EUV ∼ 1040–1042 erg s−1 at ∼55–100 eV for six galaxies, while four galaxies include large systematics in α EUV caused by stellar radiation contamination. We then compare α EUV and L EUV of these six galaxies with those predicted by the black hole (BH) accretion disk models and find that α EUV and L EUV are similar to those of the intermediate-mass BHs (IMBHs) in BH accretion disk models, albeit with possibilities of the other scenarios. Confirming these results with a known IMBH having a mass M BH of M BH = 105.75 M ⊙, we find that four local galaxies and one z = 7.665 galaxy have ionizing spectra consistent with those of IMBHs with M BH ∼ 103–105 M ⊙.

A four parameter extension to the Clement matrix and its role in numerical software testing

The Sylvester-Kac matrix (also known as the Clement matrix) and its generalizations are of interest to researchers in diverse fields. A new parameterization of the matrix has recently been presented with closed forms for the eigenvalues and Oste and Van der Jeugt (2017) have proposed their family of matrices as a source of test problems for numerical eigensolvers. In this article we extend their generalization by adding a further two parameters to the matrix definition and, for this new extension, we obtain closed forms for the eigenvalues, determinant and, the left and right eigenvectors. We show that, for certain values of the free parameters and relatively low order, these matrices may become very ill-conditioned w.r.t. both eigenvalues and inversion. In light of this we re-assess the numerical results presented by Oste and Van der Jeugt (2017) and propose a possible new testing role for parameterized Clement matrices.

Ghost dark energy in Tsallis and Barrow cosmology

According to the thermodynamics-gravity conjecture, any modification to the entropy expression leads to the modified cosmological field equations. Based on this, we investigate the cosmological consequences of the modified Friedmann equations when the entropy associated with the horizon is in the form of Tsallis/Barrow entropy and the dark energy (DE) component is in the form of ghost dark energy (GDE). We perform a dynamical system analysis and see that the Tisallis GDE(TGDE) and Barrow GDE(BGDE) can exhibit a correct phase space evolution for suitable range of the free parameters(0<Δ<1 for BGDE and β<3/2 for TGDE). We find that in BGDE and TGDE (with Q=3b2H(ρD+ρm)), there exist an early radiation dominated phase of expansion which is absent for GDE model in standard cosmology. It is worth mentioning that seeking an unstable phase of matter dominated in TGDE leads to a new constraint on β (<3/2). Using the resulting range of free parameters from latter step we find that both models are capable to explain the cosmic evolution from deceleration to an accelerated phase. We observe that increasing Δ (β) parameters leads to a delay in the cosmic phase transition. We conclude that BGDE and TGDE are viable cosmological models which predict a consistent phase transition of the cosmic expansion for suitable ranges of the parameters. We also calculate the squared sound speed for both models and find out that they are unstable against perturbations. Next, we proceed the statefinder analysis and see that both models are significantly distinct from ΛCDM as well as with respect to each other at past epochs, while both of them catch the standard cosmology at far future. We also explore the impacts of the non-extensive parameters on the density perturbations and find out that the pattern of density contrast (δ) evolution in TGDE/BGDE is distinct from GDE in standard cosmology.