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In this paper, we find the solutions for fractional potential Korteweg–de Vries (p-KdV) and Benjamin equations using q -homotopy analysis transform method ( q -HATM ) . The considered method is the mixture of q -homotopy analysis method and Laplace transform, and the Caputo fractional operator is considered in the present investigation. The projected solution procedure manipulates and controls the obtained results in a large admissible domain. Further, it offers a simple algorithm to adjust the convergence province of the obtained solution. To validate the q -HATM is accurate and reliable, the numerical simulations have been conducted for both equations and the outcomes are revealed through the plots and tables. Comparison between the obtained solutions with the exact solutions exhibits that, the considered method is efficient and effective in solving nonlinear problems associated with science and technology.

In this paper we consider two subclasses of bi-univalent functions defined by the Horadam polynomials. Further, we obtain coefficient estimates for the defined classes.

The object of the present paper is to derive some properties of holomorphic functions in the open unit disc which are defined by using a new generalized integral operator by applying a lemma due to Miller and Mocanu. Also we mention some interesting consequences of our main results.

The existence of man is dependant on nature. The peaceful existence is distributed either by man-made devastations or by natural damages. This study focuses on the contagious model of fractional order pine wilt disease employing the q-HATM (q-homotopy analysis transform method) to the leading fractional operator Atangana-Baleanu (AB) to arrive at better comprehension. The outgrowth are exhibited in the forms of figures and tables. At last the paper helps to analyse the practical theory and assists to predict its manner that correspond the guidelines contemplate the replica.

Every now and then, there has been natural or man-made calamities Such adversities instigate various institutions to find solutions for them The current study attempts to explore the disaster caused by the micro enemy called coronavirus for the past few months and aims at finding the solution for the system of nonlinear ordinary differential equations to which q?homotopy analysis transform method (q?HATM) has been applied to arrive at effective results In this paper, there are eight nonlinear ordinary differential equations considered and to solve them the advanced fractional operator Atangana-Baleanu (AB) fractional derivative has been applied to produce better understanding The outcomes have been presented in terms of plots Ultimately, the present study assists in examining the real-world models and aids in predicting their behavior corresponding to the parameters considered in the models

In this paper , we obtain initial coefficient bounds for functions belonging to the subclass of bi-univalent functions $$ B^{\lambda }_{\Sigma }(f,g,h;t)$$ by using the Chebyshev polynomials. We find Fekete–Szego inequalities and also seek the upper bound for the second Hankel determinant for functions belonging to this subclass.

The purpose of the present paper is to obtain some sufficient conditions for Mittag–Leffler-type Poisson distribution series which belongs to the classes $$\mathcal {G}(\lambda ,\; \delta )$$ and $$\mathcal {K}(\lambda ,\; \delta )$$ .

In this investigation, we propose to make use of the Horadam polynomials and introduce two classes of bi-univalent functions. For functions belonging to these classes, the coefficient inequalities and the Fekete–Szego inequalities are discussed. Some interesting remarks of the results presented here are also investigated.

The purpose of the present paper is to introduce a generalized hypergeometric distribution and obtain some necessary and sufficient conditions for generalized hypergeometric distribution series belonging to certain classes of univalent functions associated with the conic domains. We also investigate some inclusion relations. Finally, we discuss an integral operator related to this series.

The aim of this paper is applying the Differential Transformation Method (DTM) to analyze and find the solution for the mathematical model described by the system of nonlinear ordinary differential equations which describe the epidemiology of the most threatening virus called Corona-virus later labelled as COVID-19. The behaviour of the outcomes is presented in terms of plots. Finally, the present study may help you to examine the wild class of real world models and also aid to predict their behaviour with respect to parameters considered in the model. The purpose of this study is to estimate the effectiveness of preventive measures, predicting future outbreaks and potential control strategies using the mathematical model.

Our present investigation is motivated essentially by the fact that, in Geometric Function Theory, one can find many interesting and fruitful usages of a wide variety of special functions and special polynomials. The main purpose of this article is to make use of the Horadam polynomials and the generating function, in order to introduce three new subclasses of the bi-univalent function class. For functions belonging to the defined classes, we then derive coefficient inequalities and the Fekete Szego inequalities. Some interesting observations of the results presented here are also discussed. We also provide relevant connections of our results with those considered in earlier investigations., 14 pages

Systematic construction of fractional ordinary differential equations [FODEs] has gained much attention nowadays research because dimensional homogeneity plays a major role in mathematical modeling. In order to keep up the dimension of the physical quantities, we need some auxiliary parameters. When we utilize auxiliary parameters, the FODE turns out to be more intricate. One of such kind of model is non-homogeneous fractional second order RLC circuit. To solve this kind of complicated FODEs, we need proficient modern analytical method. In this paper, we use two different methods, one is modern and the other is traditional, namely generalized differential transform Method (GDTM) and Laplace transform method (LTM) to obtain the analytical solution of non-homogeneous fractional second order RLC circuit. We present the solution in terms of convergent series. Though GDTM and LTM are capable to produce the exact solution of fractional RLC circuit, great strength of GDTM over LTM is that differential transform of initial conditions occupy the coefficients of first two terms in series solution so that we arrive exact solution with few iterations and also, it does not allow the noise terms while computing the coefficients. Due to this, GDTM takes less time to converge than LTM and it has been demonstrated. Furthermost, we discuss the characteristics of non-homogeneous fractional second order RLC circuit through numerical illustrations.

In this paper, we find the necessary and sufficient conditions,inclusion relations for Poisson distribution series belonging to the classes $\mathscr{S}^*(\alpha,\beta)$ and $\mathscr{C}^*(\alpha,\beta).$ Further, we consider an integral operator related to Poisson Distribution series.

The purpose of this note is to give some mistyping corrections for our published article in [1].

In this paper, we obtain initial coefficient bounds for functions belong to a subclass of analytic bi-univalent functions related to pseudo-starlike functions by using the Chebyshev polynomials and also we find Fekete-Szego inequalities for this class.

In the current work, we delineate a comprehensive class of bi-univalent functions connected with a bounded boundary variation to get the estimates of the first two Taylor–Maclaurin coefficients. In addition, certain special cases and some appealing interpretation of the results presented here are pointed out.

We define and introduce new subclasses of harmonic univalent functions by applying the concept and theory of fractional q-calculus associated with the Ruscheweyh q-differential operator in the open unit disk. Coefficient estimates and extreme points for the functions in these classes are obtained.

This paper presents numerical solutions of the linear and nonlinear Fokker–Planck partial differential equations [FPPDEs] with space and time fractional derivatives through analytical solutions. These are treated by two analytical methods, namely, fractional reduced differential transform method [FRDTM] and fractional variational iteration method [FVIM] followed by some examples. Numerical results obtained by both FRDTM and FVIM are compared with some existing methods in the literature. This comparison shows the supremacy of FRDTM over FVIM and existing methods in terms of accuracy, simplicity and reliability.

In present work, we investigate the numerical solution of time-fractional Jaulent Miodek (JM) equations with the aid of two novel techniques namely, coupled fractional reduced differential transform method (CFRDTM) and q-homotopy analysis transform method (q-HATM). The obtained solutions are presented in a series form, which are converges rapidly. In order to verify the proposed techniques are reliable and accurate, the numerical simulations have been conducted in terms of absolute error. The obtained solutions are presented graphically to ensure the applicability and validity of the considered algorithms. The results of the study reveal that, the q-HATM is computationally very effective and accurate as compared to CFRDTM to analyse fractional nonlinear coupled Jaulent Miodek equations.

In this paper we define a subclass of bi-univalent functions. Further, we find the estimates on the bounds $|a_{2}|$ and $|a_{3}|$, the Fekete-Szego inequalities and the second Hankel determinant inequality for defined class of bi-univalent functions.

Recently, in their pioneering work on the subject of bi-univalent functions, Srivastava et al. \cite{HMS-AKM-PG} actually revived the study of the coefficient problems involving bi-univalent functions. Inspired by the pioneering work of Srivastava et al. \cite{HMS-AKM-PG}, there has been triggering interest to study the coefficient problems for the different subclasses of bi-univalent functions. Motivated largely by Ali et al. \cite{Ali-Ravi-Ma-Mina-class}, Srivastava et al. \cite{HMS-AKM-PG} and G\"{u}ney et al. \cite{HOG-GMS-JS-Fib-2018} in this paper, we consider certain classes of bi-univalent functions related to shell-like curves connected with Fibonacci numbers to obtain the estimates of second, third Taylor-Maclaurin coefficients and Fekete - Szeg\"{o} inequalities. Further, certain special cases are also indicated. Some interesting remarks of the results presented here are also discussed., Comment: 18 pages

In this paper, we introduce and investigate an interesting subclass M§(∞;‚;-;') of analytic and bi-univalent functions of complex order in the open unit disk U: For func- tions belonging to the class M§(∞;‚;-;') we investigate the coe-cient estimates on the flrst two Taylor-Maclaurin coe-cients ja2j and ja3j: The results presented in this paper would generalize and improve some recent works of (1),(5),(9).

In this paper, we define two new subclass of k-uniformly starlike and convex functions of order type with varying argument of coefficients. Further, we obtain coefficient estimates, extreme points, growth and distortion bounds, radii of starlikeness, convexity and results on modified Hadamard products.

Making use of the Faber polynomial coefficient expansions to a class of meromorphic bi-univalent functions, we obtain the general coefficient estimates for such functions and study their initial coefficient bounds. The coefficient bounds presented here are new in their own kind.

In the present work, we propose to investigate the coecient estimates for certain subclasses of bi-univalent functions of Ma-Minda type. Some interesting applications of the results presented here are also discussed.

In this short communication, the recent differential transform method is proposed to compute Laplace transforms in an innovative manner. Unlike the common method of finding Laplace transforms, the method is free of integration and hence is of computational interest. A number of illustrative examples are given to show the efficiency and simplicity of the new technique.

In their pioneering work dated 2010 on the subject of bi-univalent functions, Srivastava et al. actually revived the study of the coefficient problems involving bi-univalent functions in recent years. Inspired by the pioneering work of Srivastava et al., there has been triggering interest to study the coefficient problems for many different subclasses of bi-univalent functions. Motivated largely by a number of sequels to the investigation by Srivastava et al., in this paper, we consider certain classes of bi-univalent functions to obtain the estimates of their second and third Taylor-Maclaurin coefficients. Further, certain special cases are also indicated. Some interesting remarks about the results presented here are also discussed.

In the present investigation, we obtain some subordination and superordination results involving Hadamard products for certain normalized analytic functions in the open unit disk. Relevant connections of the results, which are presented in this paper, with various other known results are also pointed out.

Making use of Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant of a subclass $\mathcal{N}_{\sigma }^{\mu}\left( \lambda ,t\right) $ of bi-univalent function class $\sigma.$, Comment: 13 pages

Motivated and stimulated especially by the work of Xu et al. (Appl Math Lett 25(6):990–994, 2012), in this paper, we introduce and discuss an interesting subclass \(\mathcal {M}^{\varphi ,\psi }_{\Sigma }(\lambda )\) of analytic and bi-univalent functions defined in the open unit disc \(\mathbb {U}\). Further, we find estimates on the coefficients \(|a_2|\) and \(|a_3|\) for functions in this subclass. Many relevant connections with known or new results are pointed out.

In this paper, we define a new subclass of k -uniformly convex functions order α type β with varying argument of coefficients and obtain coefficient estimates. Further we investigate extreme points, growth and distortion bounds, radii of starlikeness and convexity and modified Hadamard products.

In the present work, we propose to investigate the second Hankel determinant inequalities for certain class of analytic and bi-univalent functions. Some interesting applications of the results presented here are also discussed., 8 pages

We introduce the unified biunivalent function classMq,σδ,λ(γ,φ)defined based on quasi-subordination and obtained the coefficient estimates for Taylor-Maclaurin coefficients|a2|and|a3|. Several related classes of functions are also considered and connections to earlier known and new results are established.

We introduce and study a subclass ΣP(γ,k,λ,c) of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class ΣP(γ,k,λ,c) by fixing the second coefficient. Further, it is shown that the class ΣP(γ,k,λ) is closed under convex linear combination.

In the present investigation, we obtain some subordination and superordination results involving Hadamard products for certain analytic functions. Relevant connections of the results are noted and the new results are also pointed out.

In this paper, we find the necessary and sufficient conditions for functions z F ( a , b ; c ; z ) in the generalized class of β uniformly starlike and β uniformly convex functions of order α and also consequences of the results are pointed out.