Narrow Results

We investigate the existence and convergence of approximate solutions for measure driven nonlinear differential systems defined on a separable Hilbert space. The system is restricted to finite-dimensional subspaces by using projection operators. We utilize analytic semigroup theory to construct the space in which existence and convergence are studied. In particular, the approximate solutions in finite dimensions are considered, and their convergence is verified. We apply our major findings to a partial differential equation.

Results on the existence of solutions of anti-periodic type boundary value problems for singular multi-term fractional differential equations with impulse effects are established. We first transform the problem into a hybrid system, then construct a weighted Banach space and a completely continuous operator, and finally, we use the fixed point theorem in the Banach space to prove the main results. An example is given to illustrate the efficiency of the main theorems.

In this paper, we propose a non-autonomous and diffusive SIR epidemic model based on the fact that the infection rate, the removal rate and the death rate often vary in time. The explicit formulas of the basic reproduction number ${\Re }_0$ and the minimum wave speed $c^{\ast }$ are derived. Applying upper-lower solution method and Schauder’s fixed point theorem, we show that when ${\Re }_0>1$, $c>c^{\ast }$ and the diffusion rates satisfy a certain condition, a time periodic traveling wave solution exists in the model. By the method of contradiction analysis and the comparison arguments together with the properties of the spreading speed of an associated subsystem, we prove that when ${\Re }_0\le 1$ and $c>0$ or ${\Re }_0>1$ and $0<c<c^{\ast }$, the model possesses no time periodic traveling wave solutions.

The intention of this analysis is to declare and demonstrate the complete foundational theorem of existence and uniqueness for fractional conformable initial value time-delayed models incorporating time delays. The utilized proof is based on Picard’s iterative method and the fixed point theorem. Thereafter, to showcase the outcomes of our theoretical research and as an application for numerical solutions, we propose an effective numerical method based on the reproducing kernel Hilbert approximation to solve some fractional time-delayed problems. Afterward, the construction of appropriate Hilbert spaces, derivation of the kernel functions, representation of the unique delayed solution and algorithm solution steps are also discussed and utilized. Indeed, the series solution, the convergence, and the error analysis outcomes are debated and discussed. For comparative analysis, we describe our gained results in the constructed tables and figures which help us to compare the numerical solution with the exact one. Concluding notes, keynotes and future are listed in the closing portion.

In this paper, we aim to present the concept of $S_v$-asymptotically T-periodic functions taking values in a Banach space and investigate some of their properties. Also, we establish conditions under which semi linear evolution equations in a Banach space have a unique global mild $S_v$-asymptotically T-periodic solution. Then, we apply the results obtained to prove the existence and uniqueness of $S_v$-asymptotically $\omega$-periodic mild solutions to a nonautonomous semi linear differential equations.

In this paper, we use a novel approach to study the existence, uniqueness, and stability of solutions (EUSS) to a Cauchy-type problem of nonlinear fractional differential equations (FrDiEq) of variable order with infinite delay (CPNFDEVOID). Contrary to the techniques taken in the literature, which were centered on the usage of the concept of generalized intervals and the idea of piecewise constant functions, our approach is straightforward and based on a novel fractional operator that is more appropriate and demonstrates the solvability and stability of the main problem under less restrictive presumptions. The results are achieved in this paper by using Fixed Point Theory (FPT). The application, which includes an example and supporting images, concludes the paper.

We derive a multiscale model for tumor cell migration allowing to account for the receptor-mediated movement of the cells, the degradation of tissue fibers and the subsequent production of a soluble ligand whose concentration gradient then acts together with the distribution of tissue fibers as a directional cue for the cells. For this model we present a result on the local existence and uniqueness of a solution in all biologically relevant space dimensions.

In this paper, we study on the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity. The proposed technique is graceful amalgamations of Laplace transform technique with $q$-homotopy analysis scheme and fractional derivative defined with Atangana–Baleanu (AB) operator. The fixed-point hypothesis is considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to illustrate and validate the efficiency of the future technique, we consider three different cases and analyzed the projected model in terms of fractional order. Moreover, the physical behavior of the obtained solution has been captured in terms of plots for diverse fractional order, and the numerical simulation is demonstrated to ensure the exactness. The obtained results elucidate that the proposed scheme is easy to implement, highly methodical as well as accurate to analyze the behavior of coupled nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering.

The scaling exponent of a hierarchy of cities used to be regarded as a fractional. This paper investigates a newly constructed system of equation for Hepatitis B disease in sense of Atanganaa–Baleanu Caputo (ABC) fractional order derivative. The proposed approach has five distinctive quantities, namely, susceptible, acute infections, chronic infection, immunized and vaccinated populace. By applying some well-known results of fixed point theory, we find the Ulam–Hyers type stability and qualitative analysis of the candidate solution. The deterministic stability for the proposed system is also computed. We apply well-known transform due to Laplace and decomposition techniques (LADM) and Adomian polynomial for nonlinear terms for computing the series solution for the proposed model. Graphical results show that LADM is an efficient and robust method for solving nonlinear problems.

The primary goal of this paper is to study a nonlinear fuzzy fractional dynamic system (FFDS) involving a time-dependent variational inequality. We use the monotone argument and Knaster–Kuratowski–Mazurkiewicz (KKM) theorem to prove that the variational system of FFDS is solvable and its solutions become a bounded, closed and convex set. Employing this result together with Bohnenblust–Karlin fixed point theorem and Filippov implicit function, we show the existence of a mild solution to FFDS.

This paper describes the required and adequate conditions for controllability and optimal controls of Atangana–Baleanu–Caputo (ABC) neutral fractional integrodifferential equations (NFIE) with noninstantaneous impulses. Measure of noncompactness, semigroup theory, fractional calculus and fixed point techniques act as the main tools in deriving the results. An illustration is offered to enhance our theoretical outcomes.

This work is devoted to studying the transmission dynamics of CoV-2 under the effect of vaccination. The aforesaid model is considered under fractional derivative with variable order of nonsingular kernel type known as Atangan–Baleanue–Caputo (ABC). Fundamental properties of the proposed model including equilibrium points and $R_0$ are obtained by using nonlinear analysis. The existence and uniqueness of solution to the considered model are investigated via fixed point theorems due to Banach and Krasnoselskii. Also, the Ulam–Hyers (UH) approach of stability is used for the said model. Further numerical analysis is investigated by using fundamental theorems of AB fractional calculus and the iterative numerical techniques due to Adams–Bashforth. Numerical simulations are performed by using different values of fractional-variable order $\varrho (𝜗)$ for the model. The respective results are demonstrated by using real data from Saudi Arabia for graphical presentation.

A random arbitrary-order mathematical system is investigated via the global and non-singular kernel of Atangana–Baleanu in the sense of Caputo $({𝒜}{ℬ}{𝒞})$ derivative in this study where the proposed problem is divided into four general compartments for the explanation. To show the existing result, the Krasnosilkii’s theorem from the theory of fixed points is used, whereas the well-known Banach theorem is utilized in order to show that the solution is unique to the proposed problem. Furthermore, by using the idea of Hyers–Ulam (UH) stability, the generalized problem is perturbed little for the purpose of checking its stability. The numerical solution is evaluated by applying the Adams–Bashforth iterative techniques. The numerical examples derived are tested in order to illustrate the established outcomes along with the numerical simulation to demonstrate the verification of the results obtained. The dynamics of every compartment is examined on different non-integer order ${𝒷}$ and by choosing arbitrary time $t$ by the taken approximate solution employing the AB numerical technique. Ultimately, the total continuous spectrum on the dynamics of each quantity in any arbitrary order lying between any of the two natural values, namely 0 and 1, has been achieved based on the investigated analyses.

A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the fractal-fractional order derivative (FFOD) in the Caputo sense. In addition, the tools of nonlinear functional analysis are applied to prove some qualitative results including the existence theory, stability, and numerical analysis. For the recommended results of the existence theory, Banach and Krassnoselski’s fixed point theorems are used. Additionally, Hyers–Ulam (HU) concept is used to derive some results for stability analysis. Additionally, for numerical illustration of approximate solutions of various compartments of the considered model, the modified Euler method is utilized. The aforementioned results are displayed graphically for various values of fractal-fractional orders.

The classical definitions of ‘expected value’ and ‘standard deviation’ may sometimes lead to quantities which fail to represent a ‘typical’ feature of a given data set, whenever this set consists of more than one cluster. The use of the fuzzy expected value (FEV) and the clustering fuzzy expected value (CFEV) also yield central tendency and in general cannot represent a typical value of the given data. In this work a new quantity—a Most Typical Value (MTV) is defined and investigated. A given fuzzy set in Rⁿ is first clustered and replaced by a finite set of clusters. This set is then represented by a single vector—the most typical value of the set.

We construct three kinds of periodic minimal surfaces embedded in ${ℝ}^3.$ We show the existence of a $1$-parameter family of minimal surfaces invariant under the action of a translation by $2\pi ,$ which seen from a distance look like $m$ equidistant parallel planes intersecting orthogonally $k$ equidistant parallel planes, $m,k\in \mathbb{N},$$mk\ge 2.$ We also consider the case where the surfaces are asymptotic to $m\in {\mathbb{N}}^{+}$ equidistant parallel planes intersecting orthogonally infinitely many equidistant parallel planes. In this case, the minimal surfaces are doubly periodic, precisely they are invariant under the action of two orthogonal translations. Last we construct triply periodic minimal surfaces which are invariant under the action of three orthogonal translations in the case of two stacks of infinitely many equidistant parallel planes which intersect orthogonally.

The continuous increase in unemployment rates and their significant economic impact necessitate the rapid updating and modification of present models and policies implemented by governmental entities. To successfully handle the timely transmission of employment within the workforce, many contemporary models still need the incorporation of an individual’s job history. Consequently, in order to study the unemployment problem, this research presents a multi-order fractional nonlinear mathematical model that takes into account the Caputo fractional order derivative and three important variables: the number of skilled unemployed individuals, the number of employed individuals, and the number of open positions. The existence and uniqueness of the proposed model’s solution are demonstrated by using generalization of Picard fixed point theorem. The solution of the proposed model is bounded and non-negative. The reproduction number has been analyzed to determine the factors that would help create new job vacancies. The multi-order model utilizes real data to make predictions regarding the unemployed as well as the employed population for the Northern states of India (J&K, HP, Punjab, Haryana) with an average absolute error less than 21% and 3%, respectively. When compared to the actual data, the fractional order model better captures the characteristics of the unemployed population than the integer order model. The fractional-order model exhibits lower RMSE, MAE and MAPE values and higher correlation coefficient (r) value.

In this paper, we investigate a possible applicability of the newly established fractional differentiation in the field of epidemiology. To do this, we extend the model describing the Lassa hemorrhagic fever by changing the derivative with the time fractional derivative for the inclusion of memory. Detailed analysis of existence and uniqueness of exact solution is presented using the Banach fixed point theorem. Finally, some numerical simulations are shown to underpin the effectiveness of the used derivative.

In nature there is no phenomenon that is purely periodic, and this gives the idea to consider the measure pseudo almost periodic oscillation. In this paper, by employing a suitable fixed point theorem, the properties of the measure pseudo almost periodic functions and differential inequality, we investigate the existence and uniqueness of the measure pseudo almost periodic solutions for some models of Lasota–Wazewska equation with measure pseudo almost periodic coefficients and mixed delays. We suppose that the linear part has almost periodic and the nonlinear part is assumed to be measure pseudo almost periodic. Moreover, the global attractivity and the exponential stability of the measure pseudo almost periodic solutions are also considered for the system. As application, an illustrative numerical example is given to demonstrate the effectiveness of the obtained results.

In this paper, we analyze the dynamical behavior of fish farm model related to Atangana–Baleanu derivative of arbitrary order. The model is constituted with the group of nonlinear differential equations having nutrients, fish and mussel. We have included discrete kind gestational delay of fish. The solution of fish farm model is determined by employing homotopy analysis transforms method (HATM). Existence of and uniqueness of solution are studied through Picard–Lindelof approach. The influence of order of new non-integer order derivative on nutrients, fish and mussel is discussed. The complete study reveals that the outer food supplies manage the behavior of the model. Moreover, to show the outcomes of the study, some numerical results are demonstrated through graphs.