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The computation of the clique number of a graph is a fundamental problem in graph theory, which has many applications in computational chemistry, bioinformatics, computer, and social networking. A subset [Formula: see text] of non-negative integers [Formula: see text] is called a numerical semigroup if it is a submonoid of [Formula: see text] and has a finite complement in [Formula: see text]. The graph associated with numerical semigroup [Formula: see text] is denoted by [Formula: see text] and is defined by the vertex set [Formula: see text] and the edge set [Formula: see text]. In this article, we compute the clique number and the minimum degree of those graphs, which can be associated with symmetric numerical semigroups of embedding dimension 2. Moreover, on this basis, we give some bounds for the atom-bond sum-connective index of graphs [Formula: see text] in terms of the harmonic index, the first Zagreb index, the sum-connectivity index, the maximum degree, the minimum degree, the chromatic number, and the clique number.

A subset Δ of non-negative integers [Formula: see text] is called a numerical semigroup if it is a submonoid of [Formula: see text] and has a finite complement in [Formula: see text]. A graph [Formula: see text] is called a [Formula: see text]-graph if there exists a numerical semigroup Δ with multiplicity α and embedding dimension β such that [Formula: see text] and [Formula: see text]. In this article, we compute the [Formula: see text]-graphs for irreducible and Arf numerical semigroups having a metric dimension of 2. It is proved that if Δ be an irreducible and arf numerical semigroup then there are exactly 2 and 8 non-isomorphic [Formula: see text]-graphs respectively, whose metric dimension is 2.

In this manuscript, we investigate the (k,φ)-Hilfer Langevin fractional coupled system having multi point boundary conditions and fractional integrals. This study is crucial as it addresses the complexities of (k,φ)-Hilfer Langevin coupled system with multi point conditions, which have applications in many scientific and engineering fields. Unlike previous research, our work introduces novel techniques for analyzing this system, leading to more accurate and comprehensive results. The major findings include new existence and uniqueness theorems, along with improved stability conditions, which enhance the understanding and potential applications of these systems. We demonstrate the existence, uniqueness and different kinds of Ulam stability for our suggested model. We prove the uniqueness of the solution by utilizing Banach contraction mapping principle. The existence of solution is studied by using Krasnoselskii's fixed point theorem. We also explore the different types of Ulam stability under the specific conditions. We presented an example at the end to support our main results.

The study of holomorphic functions has been recently extended through the application of diverse techniques, among which quantum calculus stands out due to its wide-ranging applications across various scientific disciplines. In this context, we introduce a novel q-differential operator defined via the generalized binomial series, which leads to the derivation of new classes of quantum-convex (q-convex) functions. Several specific instances within these classes were explored in detail. Consequently, the boundary values of the Hankel determinants associated with these functions were analyzed. All graphical representations and computational analyses were performed using Mathematica 12.0. • These classes are defined by utilizing a new q-differential operator. • The coefficient values |ai|(i=2,3,4) are investigated. • Toeplitz determinants, such as the second T2(2) and the third T3(1) order inequalities, are calculated.

Damage to bones resulting from trauma and tumors poses a significant challenge to human health. Consequently, current research in bone damage healing centers on developing three-dimensional (3D) scaffolding materials that facilitate and enhance the regeneration of fractured bone tissues. In this context, the careful selection of materials and preparation processes is essential for creating demanding scaffolds for bone tissue engineering. This is done to optimize the regeneration of fractured bones. This study comprehensively analyses the latest scientific advancements and difficulties in developing scaffolds for bone tissue creation. Initially, we clarified the composition and process by which bone tissue repairs itself. The review summarizes the primary uses of materials, both inorganic and organic, in scaffolds for bone tissue engineering. In addition, we present a comprehensive study of the most recent advancements in the mainstream techniques used to prepare scaffolds for bone tissue engineering. We also examine the distinct advantages of each method in great detail. This article thoroughly examines potential paths and obstacles in bone tissue engineering scaffolds for clinical applications.

Recent research has extensively explored classes of starlike and convex functions across various domains. This study introduces a novel class of symmetric starlike functions with respect to symmetric points and associated with the balloon shape domain. We establish the explicit representation of all functions in this class. We determine the sharp bounds for the initial four coefficients, the sharp Fekete-Szegö inequality, and the sharp bound for the second Hankel determinant for every function in the newly defined class. Furthermore, we present the new findings on the inverse and logarithmic coefficients sharp bounds for all functions belonging to this class.

Abstract Chemical graph theory has made a significant contribution to understand the chemical compound properties in the modern era of chemical science. At present, calculation of the topological indices is one of most important area of research in the field of chemical graph theory. Cyclodecane is a cyclic hydrocarbon with the chemical formula $$C_{10}H_{20}$$ C 10 H 20 . It consists of a ring of ten carbon atoms bonded together in a cyclical structure. Cyclodecane chains can be part of larger molecules or polymers, where multiple cyclodecane rings are connected together. These molecules can have various applications in chemistry, materials science, and pharmaceuticals. This article aims to determine expected values of some connectivity based topological indices of random cyclodecane chains, containing saturated hydrocarbons with at least two rings. It also compares these descriptors using explicit formulae, numerical tables and present graphical profiles of these comparisons.

Abstract It is mentioned that understanding linear and non-linear thermo-elasticity systems is important for understanding temperature, elasticity, stresses, and thermal conductivity. One of the most crucial aspects of the current research is the solution to these systems. The fractional form of several thermo-elastic systems is explored, and elegant solutions are provided. The solutions of fractional and integer thermo-elastic systems are further discussed using tables and diagrams. The closed contact between the LRPSM and exact solutions is displayed in the graphs. Plotting fractional problem solutions demonstrates their convergence towards integer-order problem solutions for suitable modeling. The tables confirm that greater precision is rapidly attained as the terms of the derived series solution increase. The faster convergence and stability of the suggested method support its modification for other fractional non-linear complex systems in nature.

Abstract This article comprises the study of class S E ∗ $\mathcal{S}_{E}^{\ast }$ that represents the class of normalized analytic functions f satisfying ς f ′ ( z ) / f ( ς ) ≺ sec h ( ς ) ${\varsigma \mathsf{f}}^{\prime }(z)/\mathsf{f}( {\varsigma })\prec \sec h ( \varsigma ) $ . The geometry of functions of class S E ∗ $\mathcal{S}_{E}^{\ast }$ is star-shaped, which is confined in the symmetric domain of a secant hyperbolic function. We find sharp coefficient results and sharp Hankel determinants of order two and three for functions in the class S E ∗ $\mathcal{S}_{E}^{\ast }$ . We also investigate the same sharp results for inverse coefficients.

Abstract Cyclooctane is a cycloalkane consisting of carbon and hydrogen atoms arranged in a closed ring structure. Cyclooctane chains can be found in various organic compounds and are significant in the field of organic chemistry due to their diverse reactivity and properties. The atom-bond connectivity index ( $$\mathcal{A}\mathcal{B}\mathcal{C}$$ A B C ), the geometric-arithmetic index ( $$\mathcal{G}\mathcal{A}$$ G A ), the arithmetic–geometric index ( $$\mathcal{A}\mathcal{G}$$ A G ) and the forgotten index ( $$\mathcal{F}$$ F ) are four well-studied molecular descriptors that have found applications in QSPR and QSAR studies. These topological descriptors have shown significant correlations with different physiochemical properties of octane isomers. In this work, the expected values of four degree based topological descriptors for random cyclooctane chains are calculated. An analytical comparison is given between the expected values of $$\mathcal{A}\mathcal{B}\mathcal{C}$$ A B C , $$\mathcal{G}\mathcal{A}$$ G A , $$\mathcal{A}\mathcal{G}$$ A G , and $$\mathcal{F}$$ F indices of random cyclooctane chains.

Abstract Two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions, are introduced and studied in this article. Under a special choice of the parameter involved, these two classes of Bazilevič functions reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root functions. Using the Faber polynomial expansion (FPE) technique, we find the general coefficient bounds for the functions belonging to each of these classes. We also find bounds for the initial coefficients for bi-Bazilevič functions and demonstrate how unexpectedly these initial coefficients behave in relation to the square-root functions and the Fibonacci-number series.

Abstract The analysis of peristaltic-ciliary transport in the human female fallopian tube, specifically in relation to the growing embryo, is a matter of considerable physiological importance. This paper proposes a biomechanical model that incorporates a finite permeable tube consisting of two layers, where the Jeffrey fluid model characterizes the viscoelastic properties of the growing embryo and continuously secreting fluid. Jeffrey fluid entering with some negative pressure gradient forms the core fluid layer while continuously secreting Jeffrey fluid forms the peripheral fluid layer. The resulting partial differential equations are solved for closed-form solutions after employing the assumption of long wavelength. The analysis delineated that increasing the constant secretion velocity, Darcy number, and Reynolds number leads to a decrease in the appropriate residue time of the core fluid layer and a reduction in the size of the secreting fluid bolus in the peripheral fluid layer. Eventually, the boluses completely disappear when the constant secretion velocity exceeds 3.0 Progesterone ( $$P_4$$ P 4 ) and estradiol ( $$E_2$$ E 2 ) directly regulate the transportation of the growing embryo, while luteinizing hormone (LH) and follicle-stimulating hormone (FSH), prolactin, anti-mullerian hormone (AMH), and thyroid-stimulating hormone (TSH) have an indirect effects. Based on the number and size of blastomeres, the percentage of fragmentation, and the presence of multinucleated blastomeres two groups were formed in an in vitro experiment. Out of 50 patients, 26 (76.5%) were pregnant in a group of the good quality embryos, and only 8 (23.5%) were in a group of the bad quality embryos. The transport of growing embryo in the human fallopian tube and preimplantation development of human embryos in in vitro are constraint by baseline hormones FSH, LH, prolactin, $$E_{2}$$ E 2 , AMH, and TSH.

This article is related to the utilization of the Laplace Adomian Decomposition Method (LADM) to analyze the solution of a fractional order system under different operators and normalized functions. In this connection, the three new and updated operators i.e., Caputo, Caputo–Fabrizio, and Atangana–Baleano are used to define the fractional order derivative with the help of four different normalization functions. It is observed that, among the four normalized functions, the fourth kind of normalized function is the most suitable function for the above-mentioned operators. The LADM series solutions are obtained for both integer and fractional orders, implying that the proposed approach has a greater accuracy and convergence rate. Figures and Tables are constructed to verify the validity and efficiency of the suggested technique. LADM simulations confirmed, that the more iterations of LADM reduced the associated absolute error. Moreover, fractional order solutions are obtained which are convergent towards integer order solutions.

Cyclooctane is classified as a cycloalkane, characterized by the chemical formula C8H16. It consists of a closed ring structure composed of eight carbon atoms and sixteen hydrogen atoms. A cyclooctane chain typically refers to a series of cyclooctane molecules linked together. Cyclooctane and its derivatives find various applications in chemistry, materials science, and industry. Topological indices are numerical values associated with the molecular graph of a chemical compound, predicting certain physical or chemical properties. In this study, we calculated the expected values of degree-based and neighborhood degree-based topological descriptors for random cyclooctane chains. A comparison of these topological indices’ expected values is presented at the end.

Tuberculosis (TB) is one of the most contagious diseases that has a greater mortality rate than HIV/AIDS and the cases of TB are feared to rise as a repercussion of the COVID-19 pandemic. The pharmaceutical industry is constantly looking for ways to improve drug design processes in order to combat the growth of infections and cure newly identified syndromes or genetically based dysfunctions with the help of QSPR models. QSPR models are mathematical tools that establish relationships between a molecular structure and its physicochemical attributes using structural properties. Topological indices are such properties that are generated from the molecular graph without any empirically derived measurements. This work focuses on developing a QSPR model using distance-based topological indices for anti-tuberculosis medications and their diverse physicochemical features.

In this research article, we introduce new family RG of holomorphic functions, which is related to the generalized bounded turning and generating functions of Gregory coefficients. Leveraging the concept of functions with positive real parts, we acquire the first five coefficients for the functions belonging to this newly defined family, demonstrating their sharpness. Furthermore, we find the third Hankel determinant for functions in the class RG. Moreover, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the under-considered class RG are estimated.

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies.

The ability of the Maximum Power Point Tracking (MPPT) technology to prevent losses by stabilizing power fluctuations during severe weather conditions is critical in improving photovoltaic power generation systems. Overall system stability is improved by carefully tracing the maximum power point (MPP). This research focuses on improving MPPT performance in solar systems by employing the ''Fuzzy Logic'' control method. The simulation, which is run in MATLAB/Simulink, includes a detailed model of the entire system. The primary circuit is designed with a DC-DC Boost architecture and a single MOSFET transistor. The Fuzzy Logic Controller (FLC) unit in MATLAB/Simulink generates an output variable led by two input variables via the Fuzzy Logic Controller unit. The Fuzzy Logic Controller (FLC) unit generates an output variable led by two input variables in MATLAB/Simulink. The simulation model, called ''Fuzzy Disturbance,'' combines Perturb & Observe with Fuzzy Logic and runs for 3 s. The results show an essential increase in system efficiency to 97%. The simulation results show that the suggested approach tracks MPP, reduces output power fluctuations, and improves system efficiency.

In theoretical chemistry, topological indices are commonly employed to model the physico-chemical properties of chemical compounds. Mathematicians frequently use Zagreb indices to calculate a chemical compound's strain energy, melting point, boiling temperature, distortion, and stability. The current global pandemic caused by the new SARS-CoV-2, also known as COVID-19, is a significant public health concern. Various therapy modalities are advised. The issue has become worse since there hasn't been enough counseling. Researchers are looking at compounds that might be used as SARS and MERS therapies based on earlier studies. In several quantitative structure-property-activity relationships (QSPR and QSAR) studies, a variety of physiochemical properties are successfully represented by topological indices, a sort of molecular descriptor that just specifies numerical values connected to a substance's molecular structure. This study investigates several irregularity-based topological indices for various antiviral medicines, depending on the degree of irregularity. In order to evaluate the effectiveness of the generated topological indices, a QSPR was also carried out using the indicated pharmaceuticals, the various topological indices, and the various physiochemical features of these antiviral medicines. The acquired results show a substantial association between the topological indices being studied by the curve-fitting approach and the physiochemical properties of possible antiviral medicines.

The well-known Chaffee–Infante reaction hierarchy is examined in this article along with its reaction–diffusion coupling. It has numerous variety of applications in modern sciences, such as electromagnetic wave fields, fluid dynamics, high-energy physics, ion-acoustic waves in plasma physics, coastal engineering, and optical fibres. The physical processes of mass transfer and particle diffusion might be expressed in this way. The Lie invariance criteria is taken into consideration while we determine the symmetry generators. The suggested approach produces the six dimensional Lie algebra, where translation symmetries in space and time are associated to mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. Additionally, similarity reductions are performed, and the optimal system of the sub-algebra should be quantified. There are an enormous number of exact solutions can construct for the traveling waves when the governing system is transformed into ordinary differential equations using the similarity transformation technique. The power series approach is also utilized for ordinary differential equations to obtain closed-form analytical solutions for the proposed diffusive coupled system. The stability of the model under the limitations is ensured by the modulation instability analysis. The reaction diffusion hierarchy’s conserved vectors are calculated using multiplier methods using Lie Backlund symmetries. The acquired results are presented graphically in 2-D and 3-D to demonstrate the wave propagation behavior.

The study of statistical distributions in a complex variable is one of the most vibrant areas of research. The complex analogue of many distributions has been studied. This article introduces the complex analogue of Binomial distribution and Negative Binomial distribution and studies certain geometrical properties of these analogues. It comprises the study of analytic functions defined by using the complex versions of Binomial distribution and Negative Binomial distribution functions. It includes the problems of finding the lower bounds of real parts of certain ratios of partial sums to the infinite series sums for these analytic functions. Such lower bounds are determined for the said analytic functions, for their first derivatives and for the Alexander transformation of these analytic functions.

It is generally considered that fractal-fractional order derivative operators are highly sophisticated mathematical tools that can be applied in a variety of physics and engineering situations to obtain real solutions. By using fractal-fractional derivatives, we can simultaneously investigate fractional order and fractal dimension. Due to extensive applications of fractal-fractional derivatives, in the present article fractal-fractional order model of non-linear Couple stress nanofluid has been analyzed. The homogenous mixture of base fluid and nanoparticles has been formed by the uniform dispersion of cadmium telluride nanoparticles in mineral transformer oil. Primarily, the classical mathematical model has been formulated via relative constitutive equations and then generalized by using fractal-fractional derivative operator. This model has been numerically solved using Crank-Nicolson technique. Using numerical solutions, various graphs are plotted to analyze how physical parameters alter Couple stress nanofluid rheology. As can be seen from the graphical study, couple stress slows down fluid velocity. Adding cadmium telluride nanoparticles to transformer oil increased its efficacy by 15.27%.

In the current note, we broaden the utilization of a new and efficient analytical computational scheme, approximate analytical method for obtaining the solutions of fractional-order Fokker-Planck equations. The approximate solution is obtained by decomposition technique along with the property of Riemann-Liouuille fractional partial integral operator. The Caputo-Riemann operator property for fractional-order partial differential equations is calculated through the utilization of the provided initial source. This analytical scheme generates the series form solution which is fast convergent to the exact solutions. The obtained results have shown that the new technique for analytical solutions is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.

In this study, we begin by examining the τ-fractional differintegral operator and proceed to establish a novel subclass in the open unit disk E. The determination of the nth coefficient bound for functions within this recently established class is accomplished by the use of the Faber polynomial expansion approach. Additionally, we examine the behavior of the initial coefficients of bi-close-to-convex functions defined by the τ-fractional differintegral operator, which may exhibit unexpected reactions. We established connections between our current research and prior studies in order to validate our significant findings.

In this research, we combine ideas from geometric function theory and fuzzy set theory. We define a new operator Dτ−λLα,ζm:A→A of analytic functions in the open unit disc Δ with the help of the Riemann–Liouville fractional integral operator, the linear combination of the Noor integral operator, and the generalized Sălăgean differential operator. Further, we use this newly defined operator Dτ−λLα,ζm together with a fuzzy set, and we next define a new class of analytic functions denoted by Rϝζ(m,α,δ). Several innovative results are found using the concept of fuzzy differential subordination for the functions belonging to this newly defined class, Rϝζ(m,α,δ). The study includes examples that demonstrate the application of the fundamental theorems and corollaries.

In this study, we apply q-symmetric calculus operator theory and investigate a generalized symmetric Sălăgean q-differential operator for harmonic functions in an open unit disk. We consider a newly defined operator and establish new subclasses of harmonic functions in complex order. We determine the sharp results, such as the sufficient necessary coefficient bounds, the extreme of closed convex hulls, and the distortion theorems for a new family of harmonic functions. Further, we illustrate how we connect the findings of previous studies and the results of this article.

In this paper, we find values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals associated with r-fold bristled graphs of ladder graphs, circular ladder graphs, some king’s graphs, and circular king’s graphs.

In this research, we define a few subclasses of analytic functions which are connected to sine functions and the cardioid domain in the unit disk. We investigate initial coefficient bounds, the Fekete–Szego problem and second and third Hankel determinants for the functions f belonging to these newly defined classes. We also define the class of m-fold symmetric functions related with the sine function and then investigate the bounds of the third Hankel determinant for twofold symmetric and threefold symmetric functions.

This article extends the study of q-versions of analytic functions by introducing and studying the association of starlike functions with trigonometric cosine functions, both defined in their q-versions. Certain coefficient inequalities like coefficient bounds, Zalcman inequalities, and both Hankel and Toeplitz determinants for the new version of starlike functions are investigated. It is worth mentioning that most of the determined inequalities are sharp with the support of relevant extremal functions.

This paper presents a geometric approach to the problems in differential subordination theory. The necessary conditions for a function to be in various subfamilies of the class of starlike functions and the class of Carathéodory functions are studied, respectively. Further, several consequences of the findings are derived.

A complex picture fuzzy set (Com-PFS) is a motivating tool for more precisely interpreting fuzzy notions. Recently, all extensions of complex fuzzy graphs (Com-FGs) have become a growing research topic as they handle ambiguous situations more explicitly than complex intuitionistic fuzzy graphs (Comp-IFG) and picture fuzzy graphs (PFG). The primary goal of this study is to demonstrate the foundation of Com-PFG due to the drawback of the complex neutral membership function in Com-IFG. This paper introduces the concept of Com-PFGs and explains many of the approaches to their development. A Com-PFG has three complex membership functions. We define the order, size, degree of vertex, and total degree of vertex in Com-PFGs. We elaborate on primary operations including: complement, join, union, Cartesian product, direct product, and composition of Com-PFG. Finally, we discuss its use in decision-making problems.

A new technique of the Adomian decomposition method is developed and applied in this research article to solve two-term diffusion wave and fractional telegraph equations with initial-boundary conditions. The proposed technique is used to solve problems of both fractional and integer orders of the telegraph equations. The fractional-order solutions provide useful information about the data transmission from one point to another. The solutions are obtained in the form of an infinite series, demonstrating a high rate of accuracy from fractional to integer orders of the problems. The technique’s accuracy is verified by drawing various fractional and integer order plots and tables. The fractional-order plots demonstrate that the solutions have a higher rate of accuracy, and the different dynamical behaviors of the problems are revealed as a result. It is discovered that the new Adomian decomposition method is the best option for solving initial-boundary value problems. The new approximations of each solution improve the method’s accuracy. As a result, it is suggested that the method can be applied to other problems with initial-boundary conditions.

This article investigates inequalities for certain types of strongly convex functions by applying q-h-integrals. These inequalities provide the refinements of some well-known results that hold for (α,m)- and (ℏ-m)-convex and related functions. Inequalities for q-integrals are deducible by vanishing the parameter h. Some particular cases are discussed after proving the main results.

In this paper, we study a new filtration class MFα,βμ, associated with the filtration of infinitesimal generators, by using the nonlinear fractional differential operator and study certain properties, like sharp Fekete–Szegö inequalities and filtration problems.

The present study introduces a new family of analytic functions by utilizing the q-derivative operator and the q-version of the hyperbolic tangent function. We find certain inequalities, including the coefficient bounds, second Hankel determinants, and Fekete–Szegö inequalities, for this novel family of bi-univalent functions. It is worthy of note that almost all the results are sharp, and their corresponding extremal functions are presented. In addition, some special cases are demonstrated to show the validity of our findings.

Materials made of graphyne, graphyne oxide, and graphyne quantum dots have drawn a lot of interest due to their potential uses in medicinal nanotechnology. Their remarkable physical, chemical, and mechanical qualities, which make them very desirable for a variety of prospective purposes in this area, are mostly to blame for this. In the subject of mathematical chemistry, molecular topology deals with the algebraic characterization of molecules. Molecular descriptors can examine a compound’s properties and describe its molecular topology. By evaluating these indices, researchers can predict a molecule’s behavior including its reactivity, solubility, and toxicity. Amidst the captivating realm of carbon allotropes, γ-graphyne has emerged as a mesmerizing tool, with exquisite attention due to its extraordinary electronic, optical, and mechanical attributes. Research into its possible applications across numerous scientific and technological fields has increased due to this motivated attention. The exploration of molecular descriptors for characterizing γ-graphyne is very attractive. As a result, it is crucial to investigate and predict γ-graphyne’s molecular topology in order to comprehend its physicochemical characteristics fully. In this regard, various characterizations of γ-graphyne and zigzag γ-graphyne nanoribbons, by computing and comparing distance-degree-based topological indices, leap Zagreb indices, hyper leap Zagreb indices, leap gourava indices, and hyper leap gourava indices, are investigated.

The aim of this paper is to introduce a class of starlike functions that are related to Bernoulli’s numbers of the second kind. Let φBS(ξ)=ξeξ−12=∑n=0∞ξnBn2n!, where the coefficients of Bn2 are Bernoulli numbers of the second kind. Then, we introduce a subclass of starlike functions 𝟊 such that ξ𝟊′(ξ)𝟊(ξ)≺φBS(ξ). We found out the coefficient bounds, several radii problems, structural formulas, and inclusion relations. We also found sharp Hankel determinant problems of this class.

This work is concerned with the branch of complex analysis known as geometric function theory, which has been modified for use in the study of fuzzy sets. We develop a novel operator Lα,λm:An→An in the open unit disc Δ using the Noor integral operator and the generalized Sălăgean differential operator. First, we develop fuzzy differential subordination for the operator Lα,λm and then, taking into account this operator, we define a particular fuzzy class of analytic functions in the open unit disc Δ, represented by Rϝλ(m,α,δ). Using the idea of fuzzy differential subordination, several new results are discovered that are relevant to this class. The fundamental theorems and corollaries are presented, and then examples are provided to illustrate their practical use.

The solutions to fractional differentials equations are very difficult to investigate. In particular, the solutions of fractional partial differential equations are challenging tasks for mathematicians. In the present article, an extension to this idea is presented to obtain the solutions of non-linear fractional Korteweg–de Vries equations. The solutions comparison of the proposed problems is done via two analytical procedures, which are known as the Residual power series method (RPSM) and q-HATM, respectively. The graphical and tabular analysis are presented to show the reliability and competency of the suggested techniques. The comparison has shown the greater contact between exact, RPSM, and q-HATM solutions. The fractional solutions are in good control and provide many important dynamics of the given problems.

In this present paper, we define a new operator in conjugation with the basic (or q-) calculus. We then make use of this newly defined operator and define a new class of analytic and bi-univalent functions associated with the q-derivative operator. Furthermore, we find the initial Taylor–Maclaurin coefficients for these newly defined function classes of analytic and bi-univalent functions. We also show that these bounds are sharp. The sharp second Hankel determinant is also given for this newly defined function class.

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, we define a new subclass of A, where the class A contains normalized analytic functions in the open unit disk E and is invariant or symmetric under rotation. First, using the Faber polynomial expansion (FPE) technique, we determine the lth coefficient bound for the functions contained within this class. We provide a further explanation for the first few coefficients of bi-close-to-convex functions defined by the q-fractional derivative. We also emphasize upon a few well-known outcomes of the major findings in this article.

Entropy is a thermodynamic function used in chemistry to determine the disorder and irregularities of molecules in a specific system or process. It does this by calculating the possible configurations for each molecule. It is applicable to numerous issues in biology, inorganic and organic chemistry, and other relevant fields. Metal–organic frameworks (MOFs) are a family of molecules that have piqued the curiosity of scientists in recent years. They are extensively researched due to their prospective applications and the increasing amount of information about them. Scientists are constantly discovering novel MOFs, which results in an increasing number of representations every year. Furthermore, new applications for MOFs continue to arise, illustrating the materials’ adaptability. This article investigates the characterisation of the metal–organic framework of iron(III) tetra-p-tolyl porphyrin (FeTPyP) and CoBHT (CO) lattice. By constructing these structures with degree-based indices such as the K-Banhatti, redefined Zagreb, and the atom-bond sum connectivity indices, we also employ the information function to compute entropies.

In this paper, we define a subclass of starlike functions associated with the Van der Pol numbers. For this class, we derive structural formula, radius of starlikeness of order α, strong starlikeness, and some inclusion results. We also study radii problems for various classes of analytic functions. Furthermore, we investigate some coefficient-related problems which include the sharp initial coefficient bounds and sharp bounds on Hankel determinants of order two and three.

This article comprises the study of strongly starlike functions which are defined by using the concept of subordination. The function φ defined by φ(ζ)=(1+ζ)λ, 0<λ<1 maps the open unit disk in the complex plane to a domain symmetric with respect to the real axis in the right-half plane. Using this mapping, we obtain some radius results for a family of starlike functions. It is worth noting that all the presented results are sharp.

Abstract In this note, we broaden the utilization of an efficient computational scheme called the approximate analytical method to obtain the solutions of fractional-order Navier–Stokes model. The approximate analytical solution is obtained within Liouville–Caputo operator. The analytical strategy generates the series form solution, with less computational work and fast convergence rate to the exact solutions. The obtained results have shown a simple and useful procedure to analyze complex problems in related areas of science and technology.

The significant security assessment is rising because the vast amount of data can be renewed continuously in the cloud. Cloud information is clustered and refreshed productively using a progressive clustering procedure over the data. To verify the security of a user's connection to the cloud, the development of information to the servers could be used. Thus, the trustworthiness of the information plays a vital role in determining the authority of the information. Map Reduce is used to deal with enormous volumes of information, and informational indexes are circulated on the cloud. A parallel information preparation structure is then received, allowing for the collection of current information that joins over time. Information anonymization methods are then used to achieve security and high information utility when an update occurs, resulting in less data loss and refresh time over time.

In this article, a new modification of the Adomian decomposition method is performed for the solution fractional order convection–diffusion equation with variable coefficient and initial–boundary conditions. The solutions of the suggested problems are calculated for both fractional and integer orders of the problems. The series of solutions of the problems with variable coefficients have been provided for the first time. To verify and illustrate our new technique, four numerical examples are presented and solved by using the proposed technique. The derived results are plotted, and the dynamics are shown for both fractional and integer orders of the problems. An excellent variation among the solutions at various fractional orders is observed. It is analyzed that the new technique based on the Adomian decomposition method is accurate and effective. The present method fits both the initial and boundary conditions with double approximations simultaneously, which increases the accuracy of the present method. For the first time, the present technique is used for the solutions of the problems with variable coefficients along with initial and boundary conditions. It is therefore suggested to apply the present procedure for the solutions of other problems with variable order and coefficients along with initial and boundary conditions.

In this research article the residual power series method is implemented for the solution of the Navier-stokes equations with two and three dimensions having the initial conditions. Caputo operator is used for the fractional derivative. The formulation is made in general form and then applied to the specific problems to check the validity of the suggested method. The solution of some numerical examples of the Navier-stokes equations are presented for both fractional and integer orders of the problems. The comparison of the obtained and exact solution is provided by using 2D and 3D plots of the solutions which confirmed the higher accuracy of the proposed method. Tables are drawn to show the results obtained, exact solutions and absolute error of the suggested method for each problem. It is investigated that as the number of terms increase in the series solution of the problems, the obtained solutions have the closed contact with actual solutions of each problem. From the calculated values it is cleared that the method is accurate and easy and therefore can be extended further to linear and nonlinear problems.