In this study, Volterra-Fredholm integral equation is solved by Hosoya Polynomials. The solutions obtained with these methods were compared on the figure and table. And error analysis was done. Matlab programming language has been used to obtain conclusitions, tables and error analysis within a certain algorithm.

In this paper, we deal with the wave equation with acoustic boundary conditions. The exponential stabilization is obtained by Lyapunov approach and Riemannian geometry method. We then apply our main theorem to the wave equations with memory type acoustic boundary conditions, which is not available in the literature and give an example in the end.

Mittag-Leffler functions has many applications in various areas of Physical, biological ,applied, earth Sciences and Engineering. It is used in solving problems of fractional order differential, integral and difference equations. In this paper, we aim to define the m-parameter Mittag-Leffler function, which can be reduced to various already known extensions of Mittag-Leffler function. We then, discuss its various properties like recurrence relations, differentiation formula and integral representations. We also represent the new m-parameter Mittag-Leffler function in terms of some known special functions such as Generalized hypergeometric function, Mellin Barnes integral, Wright hypergeometric function and Fox H-function. We also discuss its various integral transforms like Euler-Beta, Whittaker, Laplace and Mellin transforms. Further, fractional differential and integral operators are considered to discuss few properties of m-parameter Mittag-Leffler function. Also, we use the m-parameter Mittag-Leffler function to define a generalization of Prabhakar integral and discuss its properties. Further, relation of m-parameter Mittag Leffler function with various other functions such as exponential, trigonometric, hypergeometric and algebraic functions is obtained and represented graphically using MATHEMATICA 12.

Fangzhu, which has been lost for thousands of years, is an ancient device for water collection from air, its mechanism is unknown yet. Here we elucidate its possible surface-geometric and related physical properties by the oldest the Yin-Yang contradiction. In view of modern nanotechnology, we reveal that Fangzhu’s water-harvesting ability is obtained through a hydrophilic-hydrophobic hierarchy of the surface, mimicking spider web’s water collection, lotus or desert beetle’s water intake. The convex-concave hierarchy of Fangzhu’s textured surface enables it to have low wettability(high geometric potential) to attract water molecules from air through the nano-scale convex surface and transfer the attracted water along the concave surface to the collector. A mathematical model is established to reveal three main factors affecting its effectiveness, i.e., the air velocity, the surface temperature and surface structure. The lost technology can play an extremely important role in modern architecture, ocean engineering, transportation and others to catch water from air for everyday use.

The dynamic of fractional covid-19 epidemic model with a convex incidence rate is studied in this article. Under Caputo operator, existence and uniqueness for the solutions of the fractional covid-19 epidemic model have been ana- lyzed using xed point theorems. We study all the basic properties and results including local and global stability. We show the global stability of disease free equilibrium using the method of Lyapunov function theory while for disease endemic, we use the method of geometrical approach. Moreover, sensitivity analysis complemented by simulations are performed to determine how changes in parameters affect the dynamical behavior of the system.

Coronavirus disease 2019 (COVID-2019) is a viral disease which is declared as a pandemic by WHO. This disease is posing a global threat, and almost every country in the world is now affected by this disease. Currently, there is no vaccine for this disease and because of this containing COVID-19 is not an easy task. It is noticed that elderly people got severely affected by this disease specially in Europe. In the present paper, we propose and analyze a mathematical model for COVID-19 virus transmission by dividing whole population in old and young groups. We find disease-free equilibrium and the basic reproduction number ($R_0$). We estimate the parameter corresponding to rate of transmission and rate of detection of COVID-19 using real data from Italy and Spain by least square method. We also perform sensitivity analysis to identify the key parameters which influence the basic reproduction number and hence regulate the transmission dynamics of COVID-19. Finally, we extend our proposed model to optimal control problem to explore the best cost-effective and time-dependent control strategies that can reduce the number of infectives in a specified interval of time.

If an initial frame of vectors $\{e_i\}$ is related to a final frame of vectors $\{f_i\}$ by, in geometric algebra (GA) terms, a {\it rotor}, or in linear algebra terms, an {\it orthogonal transformation}, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or $4\times 4$ orthogonal matrix representing rotation and translation, given knowledge of initial and transformed points. In this paper we discuss methods in the literature for recovering such rotors and then outline a GA method which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of {\it characteristic multivectors} as discussed in the book by Hestenes \& Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create {\it fractional transformations} and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a ‘best fit’ rotor between these sets and compare our results with other methods.

The nano/microelectromechanical systems (N/MEMS) have been caught much attention in the past few decades for their attractive properties such as small size, high reliability, batch fabrication, and low power consumption. The dynamic oscillatory behavior of these systems is very complex due to strong nonlinearities in these systems. The basic aim of this manuscript is to investigate the nonlinear vibration property of N/MEMS oscillators by the homotopy perturbation method coupled with Laplace transform (also called as He-Laplace method in literature). A generalized N/MEMS oscillator is systematically studied, and a fairly accurate analytic solution is obtained. Three special cases for electrically actuated MEMS, multi-walled Carbon nanotubes-based MEMS, and MEMS subjected to van der Waals attraction are considered for comparison, and a good agreement with exact solutions is observed.

In the present paper we investigate the qualitative behaviour of a fractional SEIR model with general incidence rate function and time delay where the fractional derivative is defined in the Caputo sense. The basic reproduction number $\mathcal{R}_{0}$ is derived using the method of next generation matrix and we give a complete study of local stability of both free and endemic equilibrium. Using Liapunov method we prove the global stability of free and endemic equilibrium under some hypotheses on the parameters of the system. Finally to illustrate our results, we use the model to predict the first peak of the COVID-19 epidemic in Algeria.

The article is concerned with the analytical solution to the integro-differential system of balance and kinetic equations that describe the crystal growth phenomenon in a binary system for various nucleation kinetics. The effect of impurity concentration on the evolutionary behavior of crystals is shown. The nonlinear dynamics of a supercooled binary melt is studied with allowance for the withdrawal mechanism of product crystals from a metastable liquid of the crystallizer.

We present a framework to model and provide numerical evidence for compartmentalization in the yeast endoplasmic reticulum. Measurement data is collected and an optimal control problem is formulated as a regularized inverse problem. To our knowledge, this is the first attempt in the literature to introduce a PDE-constrained optimization formulation to study the kinetics of fluorescently labeled molecules in budding yeast. Optimality conditions are derived and a gradient descent algorithm allows accurate estimation of unknown key parameters in different cellular compartments. For the first time, the numerical results support the barrier index theory suggesting the presence of a physical diffusion barrier that compartmentalizes the endoplasmic reticulum by limiting protein exchange between the mother and its growing bud. We report several numerical experiments on real data and geometry, with the aim of illustrating the accuracy and efficiency of the method. Furthermore, a relationship between the size ratio of mother and bud compartments and the barrier index ratio is provided.

A system of boundary-domain integral equations (BDIEs) is obtained from the Dirichlet problem for the diffusion equation in non-homogeneous media defined on an exterior two-dimensional domain. We use a parametrix different from the one employed by in . The system of BDIEs is formulated in terms of parametrix-based surface and volume potentials whose mapping properties are analysed in weighted Sobolev spaces. The system of BDIEs is shown to be equivalent to the original boundary value problem and uniquely solvable in appropriate weighted Sobolev spaces suitable for unbounded domains.

This paper investigates the feedback stabilization of non-homogeneous delayed bilinear systems, evolving in Hilbert state space. More precisely, under observability like assumption, we prove the exponential and strong stability of the solution by using a bounded feedback control. The partial stabilization is discussed as well. The proof of the main results is based on the decomposition method. The decay estimates of the corresponding solution are obtained. Finally, some examples are presented.

This article specically deals with the asymptotic synchronization of non-identical complex dynamic fractional order networks with uncertainty. Initially, by using the Riemann-Liouville derivative, we developed a model for the general non-identical complex network, and based on the properties of fractional order calculus and the direct Lyapunov method in fractional order, we proposed that drive and response system if nonidentical complex networks ensuring asymp-totic synchronization by using neoteric control. Second, taking into account the uncertainties of non-identical complex networks in state matrices and evaluating theirrequirements forasymptotic synchronization. In addition, to explain the eectiveness of the proposed approach, two numerical simulations are given.

In this paper, we deal with the numerical approximation of the coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term. Since the equation is a nonlinear equation, the Raviart-Thomas mixed finite element method is one of the most suitable techniques to obtain the approximated solution. In this paper, we will show that using the Raviart-Thomas method the optimal convergence order of the scheme can be achieved. To that end, we prove the necessary lemmas and the main theorem. Finally, the efficiency of the method is certified by numerical examples.

In this paper, we study a class of sequential fractional differential inequalities involving Caputo fractional derivatives with different orders. The nonexistence of nontrivial global solutions is investigated in a suitable space via the test function technique and some properties of fractional integrals. Our results are supported by numerical examples.

In this paper, we study the vaporization process of a polydisperse ensemble of liquid drops on the basis of a nonlinear set of balance and kinetics equations for the particle-radius distribution function and temperature in the gaseous phase. We found an exact parametric solution to this problem using a modified time variable and the Laplace integral transform method. The distribution function of vaporizing drops as well as its moments, the temperature dynamics in gas, and the unvaporized mass of drops are found. The initial particle-radius distribution shifts to smaller particle radii with increasing the vaporization time. As this takes place, the temperature difference between the drops and gas decreases with time. It is shown that the heat of vaporization and initial total number of particles in the system substantially influence the dynamics of a polydisperse ensemble of liquid drops.

The formulation and analysis of two operator Boundary-Domain Integral Equation systems for variable coefficient mixed BVP with in two dimensional domain is discussed. To analyse the two-operator approach, we applied one of its linear versions to the mixed (Dirichlet-Neumann) BVP for a linear second-order scalar elliptic variable-coefficient PDE and reduced it to four different BDIE systems. %The two-operator BDIE systems are nonstandard systems of equations containing integral operators defined on the domain under consideration and potential type and pseudo-differential operators defined on open sub-manifolds of the boundary. Using the results as in CMN09 andAM11, a rigorous analysis of the two-operator BDIE systems is given and it is shown that they are equivalent to the mixed BVP and thus are uniquely solvable, while the corresponding boundary domain integral operators are invertible in the appropriate Sobolev-Slobodetski (Bessel potential) spaces.