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When software systems are introduced, they are typically deployed in field environments similar to those used during development and testing. However, these systems may also be used in various other locations with different environmental conditions, making it challenging to improve software reliability. Factors such as the specific operating environment and the location of bugs in the code contribute to this difficulty. In this paper, we propose a new software reliability model that accounts for the uncertainty of operating environments. We present the explicit closed-form mean value function solution for the proposed model. The model's goodness of fit is demonstrated by comparing it to the nonhomogeneous Poisson process (NHPP) model based on Weibull model, using four sets of failure data sets from software applications. The proposed model performs well under various estimation techniques, making it a versatile tool for practitioners and researchers alike. The proposed model outperforms other existing NHPP Weibull based in terms of fitting accuracy under two different methods of estimation and provides a more detailed and precise evaluation of software reliability. Additionally, sensitivity analysis shows that the parameters of the suggested distribution significantly impact the mean value function.

The influence of heat radiation on HMT, MHD on a porous stretched sheet has been explained in this study. The governing PDEs are converted to ODEs via similarity transformations. The coefficient of skin friction, rate of HMT are calculated using the MATLAB software for various parameter values. The controlling boundary layer PDEs are turned into a system of coupled nonlinear ODEs via similarity techniques, which are numerically solved using the shooting technique in conjunction with the fourth order Runge Kutta method, and then ANN is applied to them. Validation of numerical results using ANN results. We discovered that using engineering standpoints, an ANN model may produce high-efficiency estimates for heat transfer rates. The achieved R squared values of 99% for predicting Skin friction, Nusselt number and Sherwood number coefficients highlight the remarkable effectiveness of neural network models in these predictions. This efficacy not only demonstrates the accuracy of the models but also results in a significant reduction in the computational time required compared to traditional numerical methods. Furthermore, when compared to alternative numerical approaches, the current ANN model stands out for its applicability to more complex mathematical models, because its efficiency in minimizing both time and processing capacity demands in solving such problems.

Accurate estimation of the state of charge (SoC) of lithium-ion batteries is crucial for battery management systems, particularly in electric vehicle (EV) applications where real-time monitoring ensures safe and robust operation. This study introduces three advanced algorithms to estimate the SoC: deep neural network (DNN), gated recurrent unit (GRU), and long short-term memory (LSTM). The DNN, GRU, and LSTM models are trained and validated using laboratory data from a lithium-ion 18650 battery and simulation data from Matlab/Simulink for a LiCoO2 battery cell. These models are designed to account for varying temperatures during charge/discharge cycles and the effects of battery aging due to cycling. This paper is the first to estimate the SoC by a deep neural network using a variable current profile that provides the SoC curve during both the charge and discharge phases. The DNN model is implemented in Matlab/Simulink, featuring customizable activation functions, multiple hidden layers, and a variable number of neurons per layer, thus providing flexibility and robustness in the SoC estimation. This approach uniquely integrates temperature and aging effects into the input features, setting it apart from existing methodologies that typically focus only on voltage, current, and temperature. The performance of the DNN model is benchmarked against the GRU and LSTM models, demonstrating superior accuracy with a maximum error of less than 2.5%. This study highlights the effectiveness of the DNN algorithm in providing a reliable SoC estimation under diverse operating conditions, showcasing its potential for enhancing battery management in EV applications.

The primary objective of this study aims to carry out a more thorough investigation into a fractional nonlinear double dispersive equation that is used to represent wave propagation in an elastic, inhomogeneous Murnaghan’s rod. By Murnaghan’s rod, we mean the materials, which include the constitutive constant, Poisson ratio, and Lame’́s coefficient, are considered to be compressible in nature forming up the elastic rod. To solve the fractional version of Murnaghan’s rod problem, we employed β-fractional and M-Truncated fractional derivative. Regarding the extraction of polynomial and rational function solutions of the Murnaghan’s rod problem, which degenerate into several wave solutions including solitary, soliton (dromions), as well as periodic wave solutions. We employ the well-known unified and new auxiliary equation methods of nonlinear sciences. A finite series of certain functions satisfying an ordinary differential equation of first order, second degree is used to represent the projected solution. Based on the given approach, numerous types of solutions for exponential, hyperbolic, and trigonometric functions are generated. In this research study, the behavior of a dynamical planer system has been examined by giving various values to parameters and by depicting every possible situation as a phase portrait. The sensitivity analysis, where the soliton wave velocity and wave number parameters influence the water wave singularity, is demonstrated using the wave profiles of the constructed dynamical structural system. With the use of graphs, we have simulated the solitons to determine their kinds. All of solutions found in this manuscript is been confirmed through back substituting them into the original model using computational software.

In this paper, we investigate the existence-uniqueness, and Ulam Hyers stability (UHS) of solutions to a fractional-order pantograph differential equation (FOPDE) with two Caputo operators. Banach's fixed point (BFP) and Leray-alternative Schauder's are used to prove the existence- uniqueness of solutions. In addition, we discuss and demonstrate various types of Ulam-stability for our problem. Finally, an example is provided for clarity.

This paper presents a study of nonlinear waves in shallow water. The Korteweg-de Vries (KdV) equation has a canonical version based on oceanography theory, the shallow water waves in the oceans, and the internal ion-acoustic waves in plasma. Indeed, the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method (HPTM) to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness. This approach is connected with the fuzzy generalized integral transform and HPTM. Besides that, two novel results for fuzzy generalized integral transformation concerning fuzzy partial gH-derivatives are presented. Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method. Furthermore, 2D and 3D simulations depict the comparison analysis between two fractional derivative operators (Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense) under generalized gH-differentiability. The projected method (GHPTM) demonstrates a diverse spectrum of applications for dealing with nonlinear wave equations in scientific domains. The current work, as a novel use of GHPTM, demonstrates some key differences from existing similar methods.

Abstract This work explores the magneto-hydrodynamics (MHD) Jeffery–Hamel nanofluid flow between two rigid non-parallel plane walls with heat transfer by employing hybrid nanoparticles, especially Cu and Cu-Al $$_{2}$$ 2 O $$_{3}$$ 3 . Here the MHD nanofluid flow problem is extended with fuzzy volume fraction and heat transfer with diverse nanoparticles to cover the influence of thermal profiles with hybrid nanoparticles on the fuzzy velocity profiles. The nanoparticle volume fraction is described with a triangular fuzzy number ranging from 0 to $$5\%$$ 5 % . A novel double parametric form-based homotopy analysis approach is considered to study the fuzzy velocity and temperature profiles with hybrid nanoparticles in both convergent and divergent channel positions. Finally, the efficiency of the proposed method has been demonstrated by comparing it with the available results in a crisp environment for validation.

The evolution of nanofluids is important for improving the thermal conductivity of base fluids. The influence of thermal radiation and thermal stratification on the magnetohydrodynamic micropolar nanofluid flow through a shrinking sheet with a prescribed heat flux on the surface has been examined. The most important parts of this study are the effects of magnetohydrodynamic microrotation, thermal radiation, the magnetic field, and the Cattaneo-Christov heat flux model. The efficiency of nanoparticles, heat, and mass transference rates are influenced by the magnetic field pattern, the characteristics of the source of heat, thermal radiation, and the dispersion of volume fraction. The partial differentials are transformed into the set of nonlinear differential equations through boundary layer estimations and similarity substitutions and then computed with the use of a variational finite element procedure. A MATLAB code has been developed to assess parametric simulations for reduced skin friction factor, micro-rotation, fluid velocity, heat transfer rate, and thermal properties for the Glariken formulation. The temperature field declined due to increasing values of the thermal stratification parameter and the heat transfer rate accelerated. There is a strong link between the two sets of results, which shows that the finite element method used here is accurate.

As a large extension in Hamiltonian form, the system of a PT symmetric dimer of coupled nonlinear oscillators is developed. This system provides an explanation for a number of problems with Hamiltonian dynamics. Integrability is evaluated in the Painlevé sense of the system. The system reported twelve P-cases. First integrals of planar motion are constructed explicitly for each integrable case to show the Liouvillian integrability of the equations of motion. A mixture of numerical approaches is used to test the theoretical conclusions in order to identify the nature of orbits and evaluate the system’s transition from order to chaos. These techniques consist of the Poincaré Section Surface, the maximum Lyapunov Exponent, and the Smaller Alignment Index.

Abstract MHD Natural convection, which is one of the principal types of convective heat transfer in numerous research of heat exchangers and geothermal energy systems, as well as nanofluids and hybrid nanofluids. This work focuses on the investigation of Natural convective heat transfer evaluation inside a porous triangular cavity filled with silver-magnesium oxide/water hybrid nanofluid [H2O/Ag-MgO]hnf under a consistent magnetic field. The laminar and incompressible nanofluid flow is taken to account while Darcy–Forchheimer model takes account of the advection inertia effect in the porous sheet. Controlled equations of the work have been approached nondimensional and resolved by Galerkin finite element technique. The numerical analyses were carried out by varying the Darcy, Hartmann, and Rayleigh numbers, porosity, and characteristics of solid volume fraction and flow fields. Further, the findings are reported in streamlines, isotherms and Nusselt numbers. For this work, the parametric impact may be categorized into two groups. One of them has an effect on the structural factors such as triangular form and scale on the physical characteristics of the important outputs such as fluidity and thermal transfer rates. The significant findings are the parameters like Rayleigh and slightly supported by Hartmann along with Darcy number, minimally assists by solid-particle size and rotating factor as clockwise assists the cooler flow at the center and anticlockwise direction assists the warmer flow. Clear raise in heat transporting rate can be obtained for increasing solid-particle size.

A fourth order parallel splitting algorithm is proposed to solve one dimensional non-homogeneous heat equation with integral boundary conditions. we approximate the space derivative by fourth order finite difference approximation. This parallel splitting technique is combined with Simpson's 1/3 rule to tackle nonlocal part of this problem. The algorithm develop here is tested on two model problems. We conclude that our method provides better accuracy due to availability of real arithmetic.

In the present work, we analyze and investigate the behavior of an SEIS stochastic model which it is performed by a Lévy process. At the beginning, we established that the model has a unique global positive bounded solution. We demonstrate that the disease-free equilibrium point is stable with the suitable Lyapunov function under a certain condition. Moreover, we provide sufficient conditions for the stability of the model around the disease free and endemic equilibrium points. Finally, using a few simulation studies we demonstrate our theoretical results. We provide that, if the basic reproductive number is more than one, in some cases the virus could be extinguished under a certain condition.

In this work, a new strategy to derive inequalities by employing newly proposed fractional operators, known as a Hilfer generalized proportional fractional integral operator ($\widehat{\mathcal{GPFIO}}$). The presented work establishes a relationship between weighted extended \v{C}eby\v{s}ev version and P\'{o}lya-Szeg\"{o} type inequalities, which can be directly used in fractional differential equations and statistical theory. In addition, the proposed technique is also compared with the existing results. This work is important and timely for evaluating fractional operators and predicting the production of numerous real-world problems in varying nature.

Abstract 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.

An analogous version of Hermite-Hadamard-Mercer’s inequality has been established using the Katugampola fractional integral operators. The result is the generalization of the Riemann-Liouville fractional integral operator combined with the left and right sides of Hermite-Hadamard-Mercer’s involving differentiable mappings whose derivatives in the absolute values are convex. Our main deduction will provide noted existing results in the relative literature. It is shown, that under given conditions the derived integral inequalities, describing some kind of transfer processes, allow an exact solution, expressed by Bessel’s functions, q-digamma function and special bivariate formula. The motivation behind this investigation is to show that it has a potential application in chemical engineering.

Abstract 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 P $\mathcal{P}$ -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 P $\mathcal{P}$ -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 P $\mathcal{P}$ -convexity, we present two applications for the modified Bessel function and q $\mathfrak{q}$ -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

Coronavirus has become a serious global phenomenon in recent times and has negative effects on the entire world economy. In this study, a fractional mathematical model formulated in fractional conformable derivative is studied. The model hinges on the concept of mammal hosts and humans. The basic properties of the coronavirus model are investigated. The stability analysis is carried out as well as sensitivity analysis based on the reproduction number. Numerical simulation is undertaken to give impetus to the analytical results which indicate that both fractional conformable order derivative and fractional-order derivative have serious consequences in numerical result outcomes.

In computational mathematics, it is a matter of deep concern to recognize which of the given iteration schemes converges quickly with lesser error to the desired solution. Fixed point iterative schemes are constructed to be used for solving equations emerging in many fields of science and engineering. These schemes reformulate a nonlinear equation fs=0 into a fixed point equation of the form s=gs ; such application determines the solution of the original equation via the support of fixed point iterative method and is subject to existence and uniqueness. In this manuscript, we introduce a new modified family of fixed point iterative schemes for solving nonlinear equations which generalize further recursive methods as particular cases. We also prove the convergence of our suggested schemes. We also consider some of the mathematical models which are categorically nonlinear in essence to authenticate the performance and effectiveness of these schemes which can be seen as an expansion and rationalization of some existing techniques.

The Black-Scholes equation (BSe) is fascinating in the business world for predicting the performance of financial investment valuation systems. The Caputo fractional derivative (CFD) and Caputo-Fabrizio fractional derivative operators are used in this research to analyze the BSe. The Adomian decomposition method (ADM) and the new iterative transform (NIM) approach are combined alongside the Yang transform. In addition, the convergence and uniqueness results for the aforementioned framework have been calculated. The existence and uniqueness results have been established and frequently accompanied innovative aspects of the prospective system in fixed point terminologies. To provide additional insight into such concepts, a variety of illustrations and tabulations are used. Additionally, the provided techniques regulate and modify the obtained analytical results in a really productive fashion, allowing us to modify and regulate the converging domains of the series solution in a pragmatic manner.

Abstract The present work investigates the applicability and effectiveness of generalized proportional fractional integral ( GPFI $\mathcal{GPFI}$ ) operator in another sense. We aim to derive novel weighted generalizations involving a family of positive functions n ( n ∈ N $n\in \mathbb{N}$ ) for this recently proposed operator. As applications of this operator, we can generate notable outcomes for Riemann–Liouville ( RL $\mathcal{RL}$ ) fractional, generalized RL $\mathcal{RL}$ -fractional operator, conformable fractional operator, Katugampola fractional integral operator, and Hadamard fractional integral operator by changing the domain. The proposed strategy is vivid, explicit, and it can be used to derive new solutions for various fractional differential equations applied in mathematical physics. Certain remarkable consequences of the main theorems are also figured.

Abstract The present article addresses the concept of p-convex functions on fractal sets. We are able to prove a novel auxiliary result. In the application aspect, the fidelity of the local fractional is used to establish the generalization of Simpson-type inequalities for the class of functions whose local fractional derivatives in absolute values at certain powers are p-convex. The method we present is an alternative in showing the classical variants associated with generalized p-convex functions. Some parts of our results cover the classical convex functions and classical harmonically convex functions. Some novel applications in random variables, cumulative distribution functions and generalized bivariate means are obtained to ensure the correctness of the present results. The present approach is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractals in computer graphics.

Abstract Fractal analysis is one of interesting research areas of computer science and engineering, which depicts a precise description of phenomena in modeling. Visual beauty and self-similarity has made it an attractive field of research. The fractal sets are the effective tools to describe the accuracy of the inequalities for convex functions. In this paper, we employ linear fractals R α $\mathbb{R}^{\alpha }$ to investigate the ( s , m ) $(s,m)$ -convexity and relate them to derive generalized Hermite–Hadamard (HH) type inequalities and several other associated variants depending on an auxiliary result. Under this novel approach, we aim at establishing an analog with the help of local fractional integration. Meanwhile, we establish generalized Simpson-type inequalities for ( s , m ) $(s,m)$ -convex functions. The results in the frame of local fractional showed that among all comparisons, we can only see the correlation between novel strategies and the earlier consequences in generalized s-convex, generalized m-convex, and generalized convex functions. We obtain application in probability density functions and generalized special means to confirm the relevance and computational effectiveness of the considered method. Similar results in this dynamic field can also be widely applied to other types of fractals and explored similarly to what has been done in this paper.

Abstract In this article, the investigation is centered around the quantum estimates by utilizing quantum Hahn integral operator via the quantum shift operator ψ q η ( ζ ) = q ζ + ( 1 − q ) η ${}_{\eta}\psi_{\mathfrak{q}}(\zeta)=\mathfrak{q}\zeta+(1-\mathfrak{q})\eta$ , ζ ∈ [ μ , ν ] $\zeta\in[\mu,\nu]$ , η = μ + ω ( 1 − q ) $\eta=\mu+\frac{\omega}{(1-\mathfrak{q})}$ , 0 < q < 1 $0

Abstract In this article, we present some new properties of the fractional proportional derivatives of a function with respect to a certain function. We use a modified Laplace transform to find the relation between the derivatives in the Riemann–Liouville setting and the one in Caputo. In addition, we provide an integration by parts formulas related to the considered operators.

We consider a novel fractional model to investigates the (tuberculosis) TB model dynamics with two age groups of human, that is, the children and the adults. First, we formulate the model and present the basic results associated to the model. Then, using the fractional operators, Caputo and the Atangana-Baleanu and obtain a generalized model. Further, we give a novel numerical approach for the solution of the fractional model and obtain their approximate solution. We show graphical results with various values of the fractional order. A comparison of the two operators are shown graphically. The results obtained through Atangana-Baleanu operator is flexible than that of Caputo derivative. The infection in tuberculosis (TB) infected people decreases fast when decreasing the fractional order.

Presently the world is passing through a critical phase due to the prevalence of the Novel Corona virus, 2019-nCoV or COVID-19, which has been declared a pandemic by WHO. The virus transmits via droplets of saliva or discharge from the nose when an infected person coughs or sneezes. Due to the absence of vaccine, to prevent the disease, social distancing and proper quarantine of infected populations are needed. Non-resident citizens coming from several countries need to be quarantined for 14 days prior to their entrance. The same is to be applied for inter-state movements within a country. The purpose of this article is to propose mathematical models, based on quarantine with no lock down, that describe the dynamics of transmission and spread of the disease thereby proposing an effective preventive measure in the absence of vaccine.

This paper investigates the dynamics of a COVID-19 stochastic model with isolation strategy. The white noise as well as the Lévy jump perturbations are incorporated in all compartments of the suggested model. First, the existence and uniqueness of a global positive solution are proven. Next, the stochastic dynamic properties of the stochastic solution around the deterministic model equilibria are investigated. Finally, the theoretical results are reinforced by some numerical simulations.

The paper deals with the computation of solutions of fuzzy Volterra integral equations with degenerate kernel by applying a hybrid method. The proposed method is built on Laplace transform coupled with Adomian Decomposition Method; Laplace Adomian Decomposition Method abbreviated as (LADM). In the considered equation the unknown function has a solution in terms of infinite series expansion and hence LADM becomes more accurate to give the exact solution. Firstly, using the fuzzy number in term of parametric form, the fuzzy Volterra integral equation is converted to two crisp integral equations and then LADM is applied to get the exact fuzzy solutions of fuzzy linear Volterra integral equations. For the illustration, some examples of considered equations are solved to highlight the robustness, efficiency and the applicability of the developed scheme. The obtained results play an important role in developing the theory of fuzzy analytical dynamic equations.

In this work we focus on the eradication of the COVID-19 infection with the help of almost Non Pharmaceutical Interventions(NPIs), using mathematical modelling. First the basic reproduction number R0 is investigated. Then, on the basis of sensitivity test of R0, the most active/sensitive parameters are presented in detail. Non Pharmaceutical Interventions(NPIs) are applied to control the sensitive parameters. The major NPIs are, stay home (isolation), sanitizers (wash hands), Treatment of side effects of infection, like throat infection etc and face mask. These NPIs helps in mitigation and reducing the size of outbreak of the disease. Threshold condition for global stability of the disease free state is investigated.The NPI’s are used in different ratios to formulate a strategy. The results of these strategies are validated using Matlab software.

The main purpose of this work is to study the dynamics of a fractional-order Covid-19 model. An efficient computational method, which is based on the discretization of the domain and memory principle, is proposed to solve this fractional-order corona model numerically and the stability of the proposed method is also discussed. Efficiency of the proposed method is shown by listing the CPU time. It is shown that this method will work also for long-time behaviour. Numerical results and illustrative graphical simulation are given. The proposed discretization technique involves low computational cost.

The main objective of this paper is to develop a novel framework to study a new fractional operator depending on a parameter K > 0, known as the generalized K-fractional integral operator. To ensure appropriate selection and with the discussion of special cases, it is shown that the generalized K-fractional integral operator generates other operators. Meanwhile, we derived notable generalizations of the reverse Minkowski inequality and some associated variants by utilizing generalized K-fractional integrals. Moreover, two novel results correlate with this inequality, and other variants associated with generalized K-fractional integrals are established. Additionally, this newly defined integral operator has the ability to be utilized for the evaluation of many numerical problems.

The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of Caputo and Atangana–Baleanu under generalized Hukuhara differentiability, we illustrate the reliability of this scheme by obtaining fuzzy fractional Cauchy reaction–diffusion equations (CRDEs) with fuzzy initial conditions (ICs). Fractional CRDEs play a vital role in diffusion and instabilities may develop spatial phenomena such as pattern formation. By considering the fuzzy set theory, the proposed method enables the solution of the fuzzy linear CRDEs to be evaluated as a series of expressions in which the components can be efficiently identified and generating a pair of approximate solutions with the uncertainty parameter λ∈[0,1]. To demonstrate the usefulness and capabilities of the suggested methodology, several numerical examples are examined to validate convergence outcomes for the supplied problem. The simulation results reveal that the fuzzy SHPTM is a viable strategy for precisely and accurately analyzing the behavior of a proposed model.

The present investigation dealing with a hybrid technique coupled with a new iterative transform method, namely the iterative Elzaki transform method (IETM), is employed to solve the nonlinear fractional Fisher’s model. Fisher’s equation is a precise mathematical result that arose in population dynamics and genetics, specifically in chemistry. The Caputo and Antagana-Baleanu fractional derivatives in the Caputo sense are used to test the intricacies of this mechanism numerically. In order to examine the approximate findings of fractional-order Fisher’s type equations, the IETM solutions are obtained in series representation. Moreover, the stability of the approach was demonstrated using fixed point theory. Several illustrative cases are described that strongly agree with the precise solutions. Moreover, tables and graphs are included in order to conceptualize the influence of the fractional order and on the previous findings. The projected technique illustrates that only a few terms are sufficient for finding an approximate outcome, which is computationally appealing and accurate to analyze. Additionally, the offered procedure is highly robust, explicit, and viable for nonlinear fractional PDEs, but it could be generalized to other complex physical phenomena.

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

In this paper, we derive some new soliton solutions to $(2+1)$-Boiti-Leon Pempinelli equations with conformable derivative by using an expansion technique based on the Sinh-Gordon equation. The obtained solutions have different expression such as trigonometric, complex and hyperbolic functions. This powerful and simple technique can be used to investigate solutions of other nonlinear partial differential equations.

In Hilbert space, we develop a novel framework to study for two new classes of convex function depending on arbitrary non-negative function, which is called a predominating ℏ-convex function and predominating quasiconvex function, with respect to η , are presented. To ensure the symmetry of data segmentation and with the discussion of special cases, it is shown that these classes capture other classes of η -convex functions, η -quasiconvex functions, strongly ℏ-convex functions of higher-order and strongly quasiconvex functions of a higher order, etc. Meanwhile, an auxiliary result is proved in the sense of κ -fractional integral operator to generate novel variants related to the Hermite−Hadamard type for p t h -order differentiability. It is hoped that this research study will open new doors for in-depth investigation in convexity theory frameworks of a varying nature.

In the last decade, it has been proven in several research papers that, mathematical tools are rather power in describing real world problem in all fields of science, technology and engineering. Various mathematical models that examine real world problems have been studied and developed with the aim of predicting the future. Due to the wider applicability of these tools, research observes that the field of applied sciences is a rapidly growing discipline and has engaged the minds of researchers. The recently developed mathematical models bear certain kinds of complexities. Therefore new methods have been elaborated for observing the properties of intricated models accurately. Moreover, comprehensive information about the models have been found by modifiying the existing methods in literature. By means of the outstanding increase in information, such findings uncover new aspects and properties of real world problems. In addition, structural changes of models and technical improvements in practices brought out novel challenging issues. Such challenges have resulted in new and modified methods. Therefore, the studies in such fields are essential and meaningful in understanding the diverse aspects of the models. This special issue aims at addressing these interesting research matters in the field of applied and engineering sicences. The main source of the articles in this special issue were the selected papers from those presented at the Second International Conference on Computational Mathematics and Engineering Sciences (CMES2017), which was held on May 20-22, 2017, in Istanbul, Turkey. During CMES2017, several various papers related to applied and engineering sicence have been presented. In this special issue, we have received 12 manuscripts based on rigorous reviews. This special issue has greatly benefited from the cooperation among the authors, reviewers, and editors. We would like to express our sincere gratitude to Moulay Ismail University and Firat University for organizing CMES2017 Conference and all the authors for their contributions, which has made this special issue possible.

In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller.

In the present paper, the explicit solutions of some local fractional partial differential equations are constructed through the integration of local fractional Sumudu transform and homotopy perturbation such as local fractional dissipative and damped wave equations. The convergence aspect of this technique is also discussed and presented. The obtained results prove that the employed method is very simple and effective for treating analytically various kinds of problems comprising local fractional derivatives.