On Effects of a New Method for Fractional Initial Value Problems

The aim of this study is to analyze nonlinear Liouville-Caputo time-fractional problems by a new technique which is a combination of the iterative and ARA transform methods and is denoted by IAM. First, the ARA transform method and its inverse are utilized to get rid of time fractional derivative. Later, the iterative method is applied to establish the solution of the problem in infinite series form. The main advantages of this method are that it converges to analytic solution of the problem rapidly and implementation of method is easy. Finally, outcomes of the illustrative examples prove the efficiency and accuracy of the method.

Time fractional advection-dispersion model to study transportation of particles with time-memory for unsteady nonequilibrium suspension in open-channel turbulent flows

In this study, the effects of time-memory on the mixing and nonequilibrium transportation of particles in an unsteady turbulent flow are investigated. The memory effect of particles is captured through a time-fractional advection-dispersion equation rather than a traditional advection-dispersion equation. The time-fractional derivative is considered in Caputo sense which includes a power-law memory kernel that captures the power-law jumps of particles. The time-fractional model is solved using the Chebyshev collocation method. To make the solution procedure more robust three different kinds of Chebyshev polynomials are considered. The time-fractional derivative is approximated using the finite difference method at small time intervals and numerical solutions are obtained in terms of Chebyshev polynomials. The model solutions are compared with existing experimental data of traditional conditions and satisfactory results are obtained. Apart from this, the effects of time-memory are analyzed for bottom concentration and transient concentration distribution of particles. The results show that for uniform initial conditions, bottom concentration increases with time as the order of fractional derivative decreases. In the case of transient concentration, the value of concentration initially decreases when $T<1$ and thereafter increases throughout the flow depth. The effects of time-memory \textcolor{green}{are} also analyzed under steady flow conditions. Results show that under steady conditions, transient concentration is more sensitive for linear, parabolic, and parabolic-constant models \textcolor{green}{of} sediment diffusivity rather than the constant model.

Different soliton solutions to the modified equal-width wave equation with Beta-time fractional derivative via two different methods

‎In this paper‎, ‎different types of soliton solutions of the modified equal width wave (MEW) equation with beta time derivative are obtained by implementing the two different methods named as‎: ‎extended Jacobi's elliptic expansion function method and Kudryashov method‎. ‎The dark‎, ‎bright‎, ‎singular and other solitons are achieved‎. ‎The obtained soliton solutions are verified through MATHEMATICA‎. ‎At the end‎, ‎the results are also explained through graphs‎. ‎These soliton solutions suggest that these two methods are effective‎, ‎straight forward and reliable as compare to other methods‎. ‎The obtained results can be used in describing the substantial understanding of the studious structures as well as others related non-linear physical structures‎.

Effect of dual-rotation on MHD natural convection of NEPCM in a hexagonal-shaped cavity based on time-fractional ISPH method

The time-fractional derivative based on the Grunwald–Letnikove derivative of the 2D-ISPH method is applying to emulate the dual rotation on MHD natural convection in a hexagonal-shaped cavity suspended by nano-encapsulated phase change material (NEPCM). The dual rotation is performed between the inner fin and outer hexagonal-shaped cavity. The impacts of a fractional time derivative $$\alpha$$ α $$\left( {0.92 \le \alpha \le 1} \right)$$ 0.92 ≤ α ≤ 1 , Hartmann number Ha $$\left( {0 \le Ha \le 80} \right)$$ 0 ≤ H a ≤ 80 , fin length $$\left( {0.2 \le L_{Fin} \le 1} \right)$$ 0.2 ≤ L Fin ≤ 1 , Darcy parameter Da $$\left( {10^{ - 2} \le Da \le 10^{ - 4} } \right)$$ 10 - 2 ≤ D a ≤ 10 - 4 , Rayleigh number Ra $$\left( {10^{3} \le Ra \le 10^{6} } \right)$$ 10 3 ≤ R a ≤ 10 6 , fusion temperature $$\theta_{f}$$ θ f $$\left( {0.05 \le \theta_{f} \le 0.8} \right)$$ 0.05 ≤ θ f ≤ 0.8 , and solid volume fraction $$\varphi$$ φ $$\left( {0 \le \varphi \le 0.06} \right)$$ 0 ≤ φ ≤ 0.06 on the velocity field, isotherms, and mean Nusselt number $$\overline{Nu}$$ Nu ¯ are discussed. The outcomes signaled that a dual rotation of the inner fin and outer domain is affected by a time-fractional derivative. The inserted cool fin is functioning efficiently in the cooling process and adjusting the phase change zone within a hexagonal-shaped cavity. An increment in fin length augments the cooling process and changes the location of a phase change zone. A fusion temperature $$\theta_{f}$$ θ f adjusts the strength and position of a phase change zone. The highest values of $$\overline{Nu}$$ Nu ¯ are obtained when $$\alpha = 1$$ α = 1 . An expansion in Hartmann number $$Ha $$ Ha reduces the values of $$\overline{Nu}$$ Nu ¯ . Adding more concentration of nanoparticles is improving the values of $$\overline{Nu}$$ Nu ¯ .

Investigation of dynamics of SWCNTs and MWCNTs nanoparticles in blood flow using the Atangana–Baleanu time fractional derivative with ramped temperature

This analysis deals with the mixed free convection flow of nanofluid in the presence of porous space. Human blood is supposed to be a base fluid for which the heat transfer characteristics are performed by using the single- and multi-wall carbon nanotubes. The leading equations of the problem are obtained in dimensionless form by following the appropriate non-dimensional variables. The semi-analytical solution for the temperature and velocity field, the famous Atangana–Baleanu time-fractional derivative and Laplace transform techniques are utilized. The effects of different parameters are studied with interesting physical explanations. The summarized results show that the temperature and velocity profile decreases by varying the value of the fractional parameter. An increasing change in velocity is observed for the Grashof number. Moreover, the solution simulated via fractional model for velocity and temperature profile is more consistent and scalable for any value of the fractional parameter.

ANALYSIS OF TIME-FRACTIONAL KAWAHARA EQUATION UNDER MITTAG-LEFFLER POWER LAW

In this paper, we study a newly updated nonlinear fractional Kawahara equation (KE) using Atangana–Baleanu fractional operator in the sense of Caputo (ABC). To find the approximate solution, one of the famous techniques of the Laplace Adomian decomposition method (LADM) is used along with a time-fractional derivative. For evaluation, the required quantity is decomposing into small particles along with the application of Adomian polynomial to the nonlinear term. By the addition of the first few evaluating terms, the required convergent quantity is obtained. To explain the authenticity and the manageability of the procedure, few examples are present at different fractional orders both in three and two dimensions. Further, to compare the obtained results between fractional derivative and integer derivative, some graphical presentations are given. So, the newly updated version of the KE equation is analyzed in fraction operator providing the whole density of the total dynamics at any fractional value between two different integers.