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In the study of chemical graph theory, an enormous number of research analyses have confirmed that the characteristics of chemicals have a nearby connection with their atomic structure. Topological indices were the critical tools for the analysis of these chemical substances to consider the essential topology of chemical structures. Topological descriptors are the significant numerical quantities or invariant in the fields of chemical graph theory. In this study, we have studied the crystal structure of copper oxide ( Cu 2 O {{\rm{Cu}}}_{2}{\rm{O}} ) chemical graph, and further, we have calculated the ev-degree- and ve-degree-based topological indices of copper oxide chemical graph. This kind of study may be useful for understanding the atomic mechanisms of corrosion and stress–corrosion cracking of copper.

In the present study, a vertically oriented rectangular-shaped enclosure with triangular shaped double fins filled with PCM was considered as a Latent Thermal Energy Storage (LTES) system. Nine different cases including one pure PCM, three nano-PCM (having different volume fractions), two porous-PCM (having different porosity values) and three nano/porous PCM were considered to conduct the investigation. The PCM material employed for investigations was lauric acid. The copper made porous media was assumed to have porosity values of 98% and 95%. The Al 2 O 3 nano-particles with the volume fractions of 1%, 2% and 4% combined with Lauric acid were considered as nano-PCM. The results presented that the cases with nanoparticles volume fractions of 1% and 2% could improve the melting performance. However, the case with nanoparticle volume fraction of 4% deteriorated the melting performance. In the cases with individual usage of nano-PCM the best case was related to the case with the volume fraction of 1% and resulted in about 184s time-saving. It was found that individual usage of porous media could significantly improve the melting performance; however, adding nanoparticles to the cases with porous media, regardless of volume fraction of nanoparticles deteriorates the melting performance of PCM enclosure.

For a graph G , its variable sum exdeg index is defined as SEI a G = ∑ x y ∈ E G a d x + a d y , where a is a real number other than 1 and d x is the degree of a vertex x . In this paper, we characterize all trees on n vertices with first three maximum and first three minimum values of the SEI a index. Also, we determine all the trees of order n with given diameter d and having first three largest values of the SEI a index.

We establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville,k-fractional, Hadamard fractional, Katugampola fractional,$(k,s)$(k,s)-fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes.

Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

A novel approach to solve optimal control problems dealing simultaneously with fractional differential equations and time delay is proposed in this work. More precisely, a set of global radial basis functions are firstly used to approximate the states and control variables in the problem. Then, a collocation method is applied to convert the time-delay fractional optimal control problem to a nonlinear programming one. By solving the resulting challenge, the unknown coefficients of the original one will be finally obtained. In this way, the proposed strategy introduces a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The algorithm&rsquo, s performance has been analyzed for several non-trivial examples, and the obtained results have shown that this scheme is more accurate, robust, and efficient than most previous methods.

Thermophoretic particle deposition is one of the most rudimentary processes for transferring small particles over a temperature gradient, and it is vital in aero-solution and electronic engineering. In the proposed model, it is intended to explore heat and mass transfer on double-diffusive electrically non-conducting Casson fluid in the two-dimensional flow past a stretching surface with thermophoretic particle deposition. An external magnetic field is subjected to a magnetic dipole. The analysis of Casson fluid along magnetic dipole represents the ferromagnetic fluid. As a consequence of the presence of ferrite particles, the fluid has magnetic properties and is thus a non-Newtonian Casson ferrofluid. Moreover, to enclose heat and mass transfer characteristics, thermal and solutal stratifications are also employed. An eminent bvp4c technique in MATLAB is incorporated to compute these ODEs numerically. Graphical representations of important parameters and their physical behavior are extensively analyzed. The Nusselt number, thermophoretic particle deposition velocity, and drag force coefficient are also addressed graphically. The findings demonstrate that as the values of thermophoretic coefficient and thermophoretic parameter increase, the mass transfer rate diminishes. Further, with the escalating values of the ferromagnetic interaction parameter, the velocity field is reduced but the opposite trend is seen for the thermal field.

In this paper, a novel direct scheme to solve a set of time-delay fractional optimal control problems is introduced. The method firstly uses a set of Dickson polynomials as basis functions to approximate the states and control variables of the system. Next, the context of these basis functions and the use of a collocation method allow to transform the problem into a system of nonlinear algebraic equations. Finally, the unknown coefficients and control parameters in the original problem can be easily estimated by resolving the new system of equations. Given the high efficiency of the Dickson polynomials to deal with time-delay fractional systems, the proposed strategy involves a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The convergence analysis of the proposed method is presented, along with some illustrative examples which demonstrate its most relevant features.

The Resistance-Harary index of a connected graph G is defined as R H ( G ) = &sum, { u , v } &sube, V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index.