Search

In this article, the solutions of higher nonlinear partial differential equations (PDEs) with the Caputo operator are presented. The fractional PDEs are modern tools to model various phenomena more accurately. The residual power series method (RPSM) is used for the solution analysis of fractional partial differential equations (FPDEs), which has direct implementation for the solutions of fractional partial differential equations. In this work, the solutions to a few nonlinear FPDEs are handled by the proposed technique. The general and particular schemes of RPSM are constructed and implemented successfully. The fractional solutions of PDEs have provided many useful dynamics of the targeted problems. The RPSM results for both integer and fractional-order FPDEs are further explained and elaborated by using graphs and tables. It is observed that the higher accuracy of RPSM is achieved with fewer calculations. Graphs and tables for fractional-order solutions are presented, which confirm the convergence phenomena of fractional solutions toward integer order solutions of each problem. The suggested method can be extended to the solutions of other nonlinear fractional partial differential equations.

Usually, to find the analytical and numerical solution of the boundary value problems of fractional partial differential equations is not an easy task; however, the researchers devoted their sincere attempt to find the solutions of various equations by using either analytical or numerical procedures. In this article, a very accurate and prominent method is developed to find the analytical solution of hyperbolic-telegraph equations with initial and boundary conditions within the Caputo operator, which has very simple calculations. This method is called a new technique of Adomian decomposition method. The obtained results are described by plots to confirm the accuracy of the suggested technique. Plots are drawn for both fractional and integer order solutions to confirm the accuracy and validity of the proposed method. Solutions are obtained at different fractional orders to discuss the useful dynamics of the targeted problems. Moreover, the suggested technique has provided the highest accuracy with a small number of calculations. The suggested technique gives results in the form of a series of solutions with easily computable and convergent components. The method is simple and straightforward and therefore preferred for the solutions of other problems with both initial and boundary conditions.

The main goal of this inspection is to explore the heat and mass transport phenomena of a three-dimensional magnetohydrodynamic (MHD) flow of ternary hybrid nanoliquid through a porous media toward a stretching surface. Nowadays, the low thermal conductivity is the key problem for scientist and researchers in the transmission of heat processes. Therefore, in order to improve the thermal conductivity of different base liquids, the scientist and researchers are mixing numerous types of solid particles in the base fluids. That is why the authors have mixed three different types of nanoparticles such as graphene oxide, silver, and copper in a kerosene oil base liquid. The influences of Hall current and ion-slip are also considered. Furthermore, the flow behavior is analyzed under the appliance of Darcy–Forchheimer, activation energy, and chemical reaction. By using the concept of boundary layer theory, the flow equations are modeled in the form of higher order nonlinear partial differential equations (PDEs) along with convective boundary conditions. Suitable similarity transformations are used for the transformation of higher order PDEs into the higher order nonlinear ordinary differential equations (ODEs). Analytical scheme known as the homotopic scheme is utilized for the simulation of the current problem. The impacts of discrete flow parameters on the velocities, temperature, and concentration profiles of the ternary hybrid nanoliquid are inspected. The skin friction coefficients, Nusselt number, and Sherwood number of the ternary hybrid nanofluid are investigated against various flow parameters. The outcomes of the current analysis showed that primary velocity of the ternary hybrid nanoliquid is augmented via Hall current and ion-slip number, while the reverse trend is observed via porosity parameter, Darcy–Forchheimer parameter, and magnetic field parameter. On the other hand, the higher values of Hall current and magnetic parameter enhanced the secondary velocity of the ternary hybrid nanoliquid, while the secondary velocity was reduced due to the increasing ion-slip number and rotation parameter. It is found that the heat transfer rate of the ternary hybrid nanofluid is 46% greater than the silver nanofluid.

Several conjugate gradient (CG) parameters resulted in promising methods for optimization problems. However, it turns out that some of these parameters, for example, 'PRP,' 'HS,' and 'DL,' do not guarantee sufficient descent of the search direction. In this work, we introduce new spectral-like CG methods that achieve sufficient descent property independently of any line search (LSE) and for arbitrary nonnegative CG parameters. We establish the global convergence of these methods for four different parameters using Wolfe LSE. Our algorithm achieves this without regular restart and assumption of convexity regarding the objective functions. The sequences generated by our algorithm identify points that satisfy the first-order necessary condition for Pareto optimality. We conduct computational experiments to showcase the implementation and effectiveness of the proposed methods. The proposed spectral-like methods, namely nonnegative SPRP, SHZ, SDL, and SHS, exhibit superior performance based on their arrangement, outperforming HZ and SP methods in terms of the number of iterations, function evaluations, and gradient evaluations. [ABSTRACT FROM AUTHOR]

Diathermal oils have piqued the interest of several researchers in recent years due to their numerous industrial applications. The core aim of this study is the development of a novel fractal-fractional model to analyze the thermal performance of an oil-based tripartite hybrid nanofluid for channel flows. Graphene (Gr), magnesium oxide (MgO), and copper (Cu) nanoparticles are simultaneously dispersed in engine oil to obtain a tripartite hybrid nanofluid. To evaluate the impacts of the shape factor on thermal performance, five different shapes (brick, blade, spherical, platelet, and cylindrical) of nanoparticles are considered. The flow of considered fluid starts due to the constant motion of the right wall, which also encounters constant heating. Meanwhile, the left wall absorbs uniform radiation impacts. The mathematical model to explain this physical process is formulated by utilizing a fractal-fractional derivative, which involves a power-law kernel in its working. This model is exposed to joint employment of classical and fractal Laplace transforms to acquire the analytic solutions. These solutions are further used to develop expressions for skin friction coefficient and Nusselt number. Based on these quantities, heat transfer rate and shear stress are estimated to analyze augmentation in the thermal potential of engine oil and to observe variation in shear stress because of parametric effects. By conducting a comparative analysis between graphs of fractional and fractal-fractional models, it is concluded that a better elucidation of memory effects can be provided through the fractal-fractional approach. Furthermore, this work highlights that dispersing three different types of nanoparticles in engine oil makes it a more effective industrial fluid with a 9.58% higher heat-conduction capacity. This significant enhancement in the thermal performance of engine oil signifies its effectiveness for lubrication and cooling applications. [ABSTRACT FROM AUTHOR]

Mathematical approaches to structure model problems have a significant role in expanding our knowledge in our routine life circumstances. To put them into practice, the right formulation, method, systematic representation, and formulation are needed. The purpose of introducing soft graphs is to discretize these fundamental mathematical ideas, which are inherently continuous, and to provide new tools for applying mathematical analysis technology to real-world applications including imperfect and inexact data or uncertainty. Soft rough covering models briefly; SRC-Models), a novel theory that addresses uncertainty. In this present paper, we have introduced two new concepts Li-soft rough covering graphs (Li-SRCGs) and the concept of fixed point of such graphs. Furthermore, we looked into a some algebras that dealt with the fixed points of Li-SRCGs. Applications of the algebraic structures available in covering soft sets to soft graphs may reveal new facets of graph theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose an inertial forward–backward splitting method for solving the modified variational inclusion problem. The concept of the proposed method is based on Cholamjiak's method. and Khuangsatung and Kangtunyakarn's method. Cholamjiak's inertial technique is utilized in the proposed method for increased acceleration. Moreover, we demonstrate that the proposed method strongly converges under appropriate conditions and apply the proposed method to solve the image restoration problem where the images have been subjected to various obscuring processes. In our example, we use the proposed method and Khuangsatung and Kangtunyakarn's method to restore two medical images. To compare image quality, we also evaluate the signal-to-noise ratio (SNR) of the proposed method to that of Khuangsatung and Kangtunyakarn's method. [ABSTRACT FROM AUTHOR]