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The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm that checks if a given second‐order homogeneous linear differential equation can be reduced to the confluent Heun's equation by using the change of variables transformation and the exp‐product transformation. The main purpose of this paper is finding solutions in terms of the confluent Heun's functions for the given differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mathematical epidemic model with the effect of vaccination. We formulate a Susceptible-High risk-Infective-Recovered-Vaccinated (SHIRV) model in which the susceptible individuals with a higher probability of being infected (H) are selected as a separate class. We study the dynamical behavior of this model and define the basic reproductive number, R0. It is proved that the disease-free equilibrium is asymptotically stable if R0 < 1, and it is unstable if R0 1. Also, we investigate the existence and stability of the endemic equilibrium point analytically. For the system of differential equations of the SHIRV model, we give an approximating solution by using the Legendre-Ritz-Galerkin method. Finally, we study the influence of vaccination on measles and smallpox, two cases of the epidemic, using the proposed method in this paper. Numerical results showed that choosing high-risk people for vaccination can prevent them from getting infected and reduce mortality in the community. [ABSTRACT FROM AUTHOR]

The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences Vn, first encountered as a special sequence (corresponding to n=2) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for Vn, and we derive an exponential generating function of Vn expressed by Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear χ-fractional differential equations (χ-FDEs) with variable coefficients. The fractional derivative used in this work is the χ-Caputo fractional derivative sense (χ-CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

In the realm of modeling biological systems with memory, particularly those involving intricate interactions like tumor-immune responses, the utilization of multiple time delays and Caputo-type fractional-order derivatives represents a cutting-edge approach. In this research paper, we introduce a novel fractional-order model to investigate the dynamic interplay between tumors and macrophages, a key component of the immune system, while incorporating multiple time delays into our framework. Our proposed model comprises a system of three Caputo-type fractional-order differential equations, each representing distinct cell populations: tumor cells, anti-tumor cells (specifically M1-type macrophages with pro-inflammatory properties), and pro-tumor cells (M2-type macrophages with immune-suppressive characteristics). The stability of equilibria is discussed by analyzing the characteristic equations for each case, and the existence conditions for the Hopf bifurcation are obtained according to the critical values of delay parameters. Furthermore, numerical simulations are presented in order to verify the analytical results obtained for stability and Hopf-bifurcation with respect to the two-time delay parameters τ1 and τ2. The analysis shows the rich dynamics of the model according to the fractional-order parameter and the time delay parameters. [ABSTRACT FROM AUTHOR]

Invasive tree species pose a grave global threat to biodiversity, particularly impacting vulnerable grassland ecosystems. This paper presents a novel approach to simulate the invasion of ecosystems by these aggressive species. Our innovative coupled ODE-PDE model offers explicit invasion simulations, a significant leap beyond traditional integro-difference matrix models. With robust theoretical properties, including solution positivity, and drawing upon PDE theory, our model offers deeper insights into invasion behavior. Our model reveals that invasions undergo a prolonged quiescent phase before rapidly transitioning to a consistent rate of advancement after a critical point. This study not only sheds light on invasion dynamics but also charts a course for future exploration and potential model extensions. [ABSTRACT FROM AUTHOR]

This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: DCρω(ϰ)=fϰ,ω(ϰ),DCρω(ϰ)$$ {\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)&#x0003D;f\left(\mathrm{\varkappa},\omega \left(\mathrm{\varkappa}\right),{\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)\right) $$for all ϰ∈[0,T]$$ \mathrm{\varkappa}\in \left[0,T\right] $$ with the initial condition ω(0)=ω0$$ \omega (0)&#x0003D;{\omega}_0 $$, where 0<ρ<1,DCρ$$ 0<\rho <1,{\mathcal{D}}_C&#x0005E;{\rho } $$ is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution. [ABSTRACT FROM AUTHOR]

In this short note, we investigate a linear stochastic differential equation from mathematical physics, driven by parametric uncertainty. Given the Legendre differential equation with random inputs, the goal is to give a proof of a conjecture posed in a recent paper, concerning the power‐series solution in a Lebesgue sense. By using the finiteness of the moment‐generating function of the equation's coefficient, mean‐square convergence is derived. [ABSTRACT FROM AUTHOR]

This paper provides some information on the eigenvalues and eigenfunctions of some left‐definite system of first‐order differential equations subject to eigenparameter‐dependent boundary conditions. Namely, we show that the pair of solutions of the system of equations satisfying some initial conditions exists and is unique, and this pair is analytic in the spectral parameter of order 1/2. We also introduce Lagrange's formula for the left‐definite equation. Using some Prüfer angels, we investigate oscillation of zeros of eigenfunctions and asymptotics equations for the eigenvalues of the problem. Moreover, we share some ordinary and Fréchet derivatives of eigenvalues and eigenfunctions with respect to some elements of data. [ABSTRACT FROM AUTHOR]

In this paper, we study the inversion of the inner boundary data for Poisson equation in a doubly connected domain by fast Fourier ultraspherical spectral solver. The solver depends on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by an ultraspherical spectral method. Because this problem is seriously ill‐posed, the Cauchy data with noise will lead to ill‐conditioned linear system. Hence, we apply Tikhonov regularization to solve the obtained linear system and use generalized cross‐validation (GCV) criterion to select regularization parameters. The accuracy and efficiency of the proposed method are illustrated by several numerical results of different regions. [ABSTRACT FROM AUTHOR]

This paper presents a nonlinear model for analyzing the behavior of functionally graded material Euler‐Bernoulli thin‐walled beams with varying cross‐sections, while considering periodic perturbations in the rotating speed. The study focuses on an analytical exploration of the natural frequency and amplitude frequency response associated with the nonlinear parametric vibrations within this system. Initially, the governing differential equations for the transverse vibrations of the beams are derived through Hamiltonian principle. Subsequently, the perturbation method is employed to solve the established equations, and Galerkin method is adopted to determine the mode shapes and frequencies. Furthermore, this research extensively examines the influence of significant parameters on the vibration characteristics of the system. [ABSTRACT FROM AUTHOR]

The main theme of this study is to implement the Sumudu integral transform technique to solve the stability problem of linear differential equations. Another important aspect of this paper is to investigate the Ulam–Hyers and Ulam–Hyers–JRassias stability of linear differential equations by using Sumudu transform method. Further, the results are extended to the Mittag‐Leffler–Ulam–Hyers and Mittag‐Leffler–Ulam–Hyers–JRassias stability of these differential equations. As an application point of view, the Sumudu transform is used to find Ulam stabilities of differential equations arising in field‐controlled DC servo motor with position control. [ABSTRACT FROM AUTHOR]

During past decades, the study of the interaction between predator and prey species has become one of the most exciting topics in computational biology and mathematical ecology. In this paper, we aim to investigate the stability of a diseased model of susceptible, infected prey and predators around an internal steady state. To this end, the fractional derivatives based on the Mittag‐Leffler kernels in the Liouville‐Caputo concept has been taken into consideration. The existence and uniqueness of the acquired solutions to the model are also studied in this paper. In order to investigate the effects of the fractional‐order along with other existing parameters in the model, several possible scenarios have been examined. As it is seen in the proposed graphical simulations, the employed fractional operator is capable of capturing all anticipated theoretical features of the model. The numerical technique employed in this contribution is precise and efficient and can be easily adopted to investigate many fractional‐order models in biology. It is found that new proposed operators of fractional‐order can describe the real‐world phenomena even better than integer‐order differential equations because of their memory‐related properties. [ABSTRACT FROM AUTHOR]

In this paper, we establish global fractional Halanay‐type inequalities with distributed delays in differential and integral forms and consequently give a simple sufficient condition for Mittag–Leffler stability of solutions to fractional differential equations with this type of delays. Our essential tool is the submultiplicative property of the Mittag–Leffler functions which imitates the multiplicative property of the exponential functions in the classical theory of differential equations. The estimation with explicit decaying rate and amplitude constant is a faithful extension of the classical Halanay inequality. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the class of Eberlein weakly almost periodic functions in the Stepanov's sense is strictly larger than both Eberlein weak almost periodicity (E‐w.a.p.) and Stepanov's almost periodicity. Some examples are provided. We present a composition result for this class of functions, and using Banach fixed‐point theorem, we prove the existence and uniqueness of E‐w.a.p. solution for a differential equation of neutral type with nondense domain when the forcing terms are only E‐w.a.p. in the Stepanov's sense. We mention that our results can extend and improve previous several works in this direction. An application is given to illustrate the result. [ABSTRACT FROM AUTHOR]

This paper is dedicated to obtaining closed‐form solutions of linear difference equations which are asymptotically close to the self‐adjoint Euler‐type difference equation. In this sense, the equation is related to the Euler–Cauchy differential equation y′′+λ/t2y=0$$ {y}^{\prime \prime }+\lambda /{t}^2y=0 $$. Throughout the paper, we consider a system of sequences which behave asymptotically as an iterated logarithm. [ABSTRACT FROM AUTHOR]

The paper deals with a new type of inverse spectral problems for second‐order quadratic differential pencils when one of the boundary conditions involves arbitrary entire functions of the spectral parameter. Although various aspects of the inverse spectral theory for the pencils have been of a special interest during the last decades, such settings were considered before only in the particular case of a Sturm–Liouville equation. We develop an approach covering also the quadratic dependence on the spectral parameter in the differential equation, which is based on the completeness and basisness of certain functional systems. By this approach, we obtain a uniqueness theorem and an algorithm for solving the inverse problem along with sufficient properties of the mentioned systems. The presented results give a universal tool for studying a number of important specific situations, including various Hochstadt–Lieberman‐type inverse problems both on an interval and on geometrical graphs, which is illustrated as well. [ABSTRACT FROM AUTHOR]

In this paper, we study asymptotic behaviors for classical thermoelastic plate equations with the Fourier law of heat conduction in the whole space ℝn$$ {\mathrm{\mathbb{R}}}^n $$, where we introduce a reduction methodology basing on third‐order (in time) differential equations and refined Fourier analysis. We derive optimal growth estimates when n⩽3$$ n\leqslant 3 $$, bounded estimates when n=4$$ n=4 $$, and decay estimates when n⩾5$$ n\geqslant 5 $$ for the vertical displacement in the L2$$ {L}^2 $$ norm. Particularly, the new critical dimension n=4$$ n=4 $$ for distinguishing the decisive role between the plate model and the Fourier law of heat conduction is discovered. Moreover, concerning the small thermal parameter in the temperature equation, we study the singular limit problem. We not only show global (in time) convergence of the vertical displacements between thermoelastic plates and structurally damped plates but also rigorously demonstrate a new second‐order profile of the solution. Our methodology can settle several closely related problems in thermoelasticity. [ABSTRACT FROM AUTHOR]

In this paper, we consider n$$ n $$‐dimensional systems of differential equations applied to model a range of physical phenomena, where n$$ n $$ represents any positive integer. The defining characteristic that we focus on are the magnitudes of the inherent physical parameters, which are frequently of different orders of magnitude for various physical systems. This property, inherent to the given physical problems, can be exploited to reduce the problems into a set of much simpler subproblems, expanding our ability to gain physical insight in terms of the underlying simpler subproblems. These orders of magnitude motivate the choice of a specific class of transformations that can be chosen to reduce the number of required dynamic iterations in a sequence‐based approach to examining such systems. This is also useful for the development of proof techniques for proving properties of mathematical models, such as their well‐posedness. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the multiterm time‐fractional partial differential equation involving the Caputo operator associated with the Laplace operator, which includes the momentum equations of the fractional Oldroyd‐B fluid and the fractional Burgers fluid under some suitable conditions. By using several properties of multivariate Mittag–Leffler functions, the well‐posedness and the long‐time behavior for the Dirichlet problem are obtained. In addition, the uniqueness in inverse problem of determining orders of time‐fractional derivatives of the equation is proved. [ABSTRACT FROM AUTHOR]

In this paper, we study the full parabolic attraction–repulsion Keller–Segel model with a fractional diffusion in ℝn$$ {\mathbb{R}}^n $$ for n=2$$ n=2 $$ or 3. We are more interested in the question that whether the solutions exist globally or blow up in finite time, which was studied in the classical attraction‐repulsion Keller‐Segel model by Jin and Wang (J. Differential Equations, 2016) through constructing a suitable energy functional. However, for the fractional attraction–repulsion Keller–Segel model, it is challenging to find a similar energy functional to study the existence of the solutions. In the present paper, under the condition of ξγ=χα$$ \xi \gamma =\chi \alpha $$, we introduce the Lp$$ {L}_p $$‐ Lq$$ {L}_q $$ estimates of the fractional derivative modulus of the solution to carry out the first step of the problem. Armed with a priori estimate, it is sufficient for us to obtain the global existence of the solutions by the Moser–Alikakos iterative method and finally arrive at the decay estimates of the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the qualitative behavior of a class of fractional SEIR epidemic models with a more general incidence rate function and time delay to incorporate latent infected individuals. We first prove positivity and boundedness of solutions of the system. The basic reproduction number R0$$ {\mathcal{R}}_0 $$ of the model is computed using the method of next generation matrix, and we prove that if R0<1$$ {\mathcal{R}}_0<1 $$, the healthy equilibrium is locally asymptotically stable, and when R0>1$$ {\mathcal{R}}_0>1 $$, the system admits a unique endemic equilibrium which is locally asymptotically stable. Moreover, using a suitable Lyapunov function and some results about the theory of stability of differential equations of delayed fractional‐order type, we give a complete study of global stability for both healthy and endemic steady states. The model is used to describe the COVID‐19 outbreak in Algeria at its beginning in February 2020. A numerical scheme, based on Adams–Bashforth–Moulton method, is used to run the numerical simulations and shows that the number of new infected individuals will peak around late July 2020. Further, numerical simulations show that around 90% of the population in Algeria will be infected. Compared with the WHO data, our results are much more close to real data. Our model with fractional derivative and delay can then better fit the data of Algeria at the beginning of infection and before the lock and isolation measures. The model we propose is a generalization of several SEIR other models with fractional derivative and delay in literature. [ABSTRACT FROM AUTHOR]

This paper introduces the quaternion Schwarz type space, and quaternion linear canonical transform (QLCT) mapping properties are also discussed. Further, the quaternion pseudo‐differential operator (QPDO) associated with QLCT is described. Some of its characteristics, including estimates, boundedness, and integral representation in quaternion Sobolev type space, are derived. Some applications of QLCT, quaternion differential equations, are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method for American multi‐asset options under jump‐diffusion model based on the combination of the exponential time differencing (ETD) technique for the differential operator and Gauss–Hermite quadrature for the integral term. In order to simplify the computational stencil and improve characteristics of the ETD‐scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag–Leffler kernel. Firstly, we establish the existence of solutions to the studied problem by using the Banach contraction theorem. Then, by means of the Dhage fixed‐point principle, we discuss the existence of mild solutions. Finally, we study the Ulam–Hyers stability of the introduced fractional hybrid problem. [ABSTRACT FROM AUTHOR]

The well‐known Drinfeld–Sokolov–Wilson system is widely used to describe the flow of the shallow water. This paper considers the time–space fractional Drinfeld–Sokolov–Wilson system, where He's variational method together with the two‐scale transform is applied to find its solitary solutions. The numerical results have been shown through graphs to demonstrate the applicability and efficiency of our approach. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

Mathematics and its applications try to make available a simple and accurate approach to handle nonlinear equations. The present paper investigates for the first time the application of the linearizing method to determine both the approximate frequency and displacement amplitude for the third‐order differential equations. The main merit of the linearized method is a collection of simplicity and accuracy for the solution of high‐order nonlinear conservative or non‐conservative oscillators. The current work sheds light on the approximate proposition to the damped third‐order nonlinear differential equations. The fast estimated solution is simulated to the accurate solution gained through the numerical methods. It is significant to conclude that the non‐perturbative method considered here is simpler and easier to apply than the other perturbation methods. This paper enriches some existing literature. [ABSTRACT FROM AUTHOR]

We consider both primal and dual formulations of singular autonomous systems of three different types of fractional‐order differential equations. We present a comprehensive study which proves that by using the spectrum of a linear pencil, a polynomial matrix of first order, and not the fractional‐order pencil of the prime system, we will receive information for all properties for both the prime and its dual system. In addition, by using this spectrum, the solutions for all systems can be obtained by using formulas without additional computational cost. Finally, we provide examples including a computational analysis in Modelica. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of singular prey‐taxis models in a smooth bounded domain under homogeneous Neumann boundary conditions. The main challenge of analysis is the possible singularity as the prey density vanishes. Employing the technique of a priori assumption, the comparison principle of differential equations and semigroup estimates, we show that the singularity can be precluded if the intrinsic growth rate of prey is suitably large and hence obtain the existence of global classical bounded solutions. Moreover, the global stability of co‐existence and prey‐only steady states with convergence rates is established by the method of Lyapunov functionals. [ABSTRACT FROM AUTHOR]

In this paper, we present an operational matrix method to obtain numerical solutions of ψ$$ \psi $$‐Caputo fractional ordinary and partial differential equations. For this purpose, a fractional version of the Taylor theorem is presented in the framework of ψ$$ \psi $$‐fractional calculus. The method converts the underlying ordinary or partial differential equations to systems of algebraic equations. The method is accompanied by numerical examples to verify the applicability and effectiveness of the proposed method. Further, estimates of upper bounds of error for the approximations have been derived. [ABSTRACT FROM AUTHOR]

In this paper, several theorems and relations are examined by using a generalized Laplace‐type integral transform. A generalization of the harmonic oscillator in a non‐resisting and resisting medium problems, initial‐boundary problems, and integral equations is solved via this integral transform. Furthermore, the well‐known series entitled as Basel problem is obtained in a similar way. Moreover, we compare numerically classical and newly introduced by us integral transform. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following chemotaxis model with indirect signal consumption: ut=Δum−∇·(u∇v),x∈Ω,t>0,vt=Δv−vw,x∈Ω,t>0,wt=−δw+u,x∈Ω,t>0$$ \left\{\begin{array}{ccc}\hfill {u}_t& =\Delta {u}^m-\nabla \cdotp \left(u\nabla v\right),\hfill & \hfill \kern2em x\in \Omega, t>0,\\ {}\hfill {v}_t& =\Delta v- vw,\hfill & \hfill \kern2em x\in \Omega, t>0,\\ {}\hfill {w}_t& =-\delta w+u,\hfill & \hfill \kern2em x\in \Omega, t>0\end{array}\right. $$under the homogeneous Neumann boundary conditions for u$$ u $$ and v$$ v $$ in a bounded convex domain Ω⊂ℝ3$$ \Omega \subset {\mathbb{R}}^3 $$ with smooth boundary, where δ>0$$ \delta >0 $$ is a given parameter. It is shown that for each m>4/3$$ m>4/3 $$, the global weak solutions exist whenever the initial data (u0,v0,w0)$$ \left({u}_0,{v}_0,{w}_0\right) $$ are sufficiently regular satisfying u0≥0,v0≥0$$ {u}_0\ge 0,{v}_0\ge 0 $$ and w0≥0$$ {w}_0\ge 0 $$. Moreover, it is proved that the global solution (u,v,w)$$ \left(u,v,w\right) $$ converges to ū0,0,ū0δ$$ \left({\bar{u}}_0,0,\frac{{\bar{u}}_0}{\delta}\right) $$ in L∞(Ω)$$ {L}^{\infty}\left(\Omega \right) $$ as t→∞$$ t\to \infty $$. In particular, we extend the recent result of Liu–Li–Huang [Y Liu, Z Li and J Huang, Global boundedness and large time behavior of a chemotaxis system with indirect signal absorption, J. Differential Equations 269 (2020) 6365‐6399] to the nonlinear diffusion case. [ABSTRACT FROM AUTHOR]

This paper is in concern with the study of differential problems involving the ψ$$ \psi $$‐shifted fractional derivatives, where ψ$$ \psi $$ is a scaling function. Such operators can be thought of as a generalization of several fractional derivatives such as the classical Riemann–Liouville and Caputo operators, the Hadamard operators, the generalized fractional operators, and the Erdélyi–Kober operators, among others. Two types of boundary conditions are considered: the Cauchy problems and the integral boundary conditions. [ABSTRACT FROM AUTHOR]

Neural network has good self‐learning and adaptive capabilities. In this paper, a wavelet neural network is proposed to be used to solve the value problem of fractional differential equations (FDE). We construct a wavelet neural network (WNN) with the structure 1 ×N× 1 based on the wavelet function and give the conditions for the convergence of the given algorithm. This method uses the truncated power series of the solution function to transform the original differential equation into an approximate solution, then, using WNN, update the parameters, and finally get the FDE solution. Simulation results prove the validity of WNN. [ABSTRACT FROM AUTHOR]

Weyl almost automorphy is a natural generalization of Bochner almost automorphy and Stepanov almost automorphy. However, the space composed of Weyl almost automorphic functions is not a Banach space. Therefore, the results of the existence of Weyl almost automorphic solutions of differential equations are few, and the results of the existence of Weyl almost automorphic solutions of difference equations are rare. Since the study of dynamic equations on time scales can unify the study of differential equations and difference equations. Therefore, in this paper, we first propose a concept of Weyl almost automorphic functions on time scales and then take the Clifford‐valued shunt inhibitory cellular neural networks with time‐varying delays on time scales as an example of dynamic equations on time scales to study the existence and global exponential stability of their Weyl almost automorphic solutions. We also give a numerical example to illustrate the feasibility of our results. [ABSTRACT FROM AUTHOR]

This paper considers time‐fractional (2+1)$$ \left(2+1\right) $$‐dimensional Wu–Zhang nonlinear system of partial differential equation describing a long dispersive wave. An approximate analytical solution of the dispersion relation of the long wave has been obtained by the fractional reduced differential transform method (FRDTM). The effect of fractional‐order α$$ \alpha $$ on the wave profile of the solution is discussed graphically and comparing the exact solution of Wu–Zhang equation when α=1$$ \alpha =1 $$. The result shows that the present method reveals the effectiveness, efficiency, and reliability of computed mathematical results to easily solve the fractional‐order Wu–Zhang (WZ) system of differential equations. [ABSTRACT FROM AUTHOR]

The paper treats the problem of an approximate calculation of unknown probability distribution function defining an observation set. The observation set is usually used as a supervised set to train a neural network or to build functions for machine learning. In order to solve the problem, we propose applying optimal control tools. To this effect following Vapnik, we define the risk functional depending on probability distribution functions, and we construct a set of functions based on the observation set. The construction draw on using only those functions that satisfy some differential equation with suitable properties as it is usually done in optimal control theory. Next, we apply a dual dynamic programming to formulate a new optimal control problem. As a consequence, we state and prove a verification theorem for an approximation of an unknown probability distribution function defining the observation set. We estimate the difference between the approximate probability distribution function and the best probability distribution with distance defined by the risk functional. [ABSTRACT FROM AUTHOR]

When a real‐life problem is mathematically modeled by differential equations or another type of equation, there are always intrinsic phenomena that are not taken into account and can affect the behavior of such a model. For example, external forces can abruptly change the model; impulses and delay can cause a breakdown of it. Considering these intrinsic phenomena in the mathematical model makes the difference between a simple differential equation and a differential equation with impulses, delay, and nonlocal conditions. So, in this work, we consider a semilinear nonautonomous neutral differential equation under the influence of impulses, delay, and nonlocal conditions. In this paper we study the controllability of these semilinear neutral differential equations with some of these intrinsic phenomena taking into consideration. Our aim is to prove that the controllability of the associated ordinary linear differential equation is preserved under certain conditions imposed on these new disturbances. In order to achieve our objective, we apply Rothe's fixed point Theorem to prove the exact controllability of the system. Finally, our method can be extended to the evolution equation in Hilbert spaces with applications to control systems governed by PDE's equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to the implementation of fractional differential transform method (FDTM) for the evaluation of new approximations for the fractional models of the system of multi‐pantograph differential equations. Some new properties of FDTM have also been proposed for the fractional integro‐differential equations. It is noteworthy that the integro‐differential equations are studied for integers in the literature, and we have proposed the solutions of proportional delayed integro‐differential equations for fractional order. The evaluated results are verified numerically by considering some test examples of fractional integro‐differential equations and fractional models of the system of multi‐pantograph differential equations. The outcomes conclude that evaluated series solutions via FDTM promise more appropriate approximations possess very negligible error and represent the convergent approximations. Thus, evaluated series solutions are comparatively more accurate and require less computational time. [ABSTRACT FROM AUTHOR]

In the present paper, a nonlinear fractional Schrödinger integro‐differential equation is considered in a Hilbert space. Operator approach is applied on multidimensional problems with nonlinearity that deserve a studious treatment. In this paper, theorems on existence and uniqueness of a bounded solution for the abstract problem are achieved. Additionally, existence theorems are obtained for first and second orders of accuracy difference schemes of the abstract problem. Furthermore, theorems are applied on a one‐dimensional problem with nonlocal condition and a multidimensional problem with Dirichlet boundary condition. Numerical results and illustrations are presented to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study periodic and S‐asymptotically periodic solutions for fractional diffusion equations (FDE). As we all know, there is no exact periodic solution to differential equations with Caputo or Riemann‐Liouville fractional derivatives. Even so, in this paper, periodic (S‐asymptotically periodic) mild or classical solutions for FDE with Weyl‐Liouville fractional derivatives could be obtained in various fractional power spaces. In addition, a numerical simulation example and a specific example of fractional diffusion equation are given to verify the main theoretical results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of a second‐order differential equation of type Sturm‐Liouville with coefficients sign changing, and the existence of one positive solution is obtained by requiring a specific growth of the nonlinearity. In particular, this study is useful in the analysis of the dynamical performance of a class of power converter. [ABSTRACT FROM AUTHOR]

This paper is a study about a fractional‐order hybrid system of two fractional‐order differential equations. We have given the existence of solution and uniqueness of solutions, and finally, the numerical solutions for an application is presented. The existence results are based on the classical fixed‐point approach while the numerical results are obtained with the help of fractional‐order Euler's technique. In the numerical solutions, we can see that the orders of the differential equations are playing important role. We have given related literature for the readers for their detail explanation. [ABSTRACT FROM AUTHOR]

In a recent paper (Nakamura and Martinez, 2019), the classical epidemic compartmental susceptible‐infectious‐susceptible (SIS) model has been upgraded to a form which permits fluctuations in terms of the mean and the variance of infected individuals. This novel model happens to admit a Hamiltonian realization involving a constant ρ0 that is related to the reproduction number R0. In this work, we generalize ρ0 as a t‐dependent function to arrive at a non‐autonomous system of differential equations that we call SISt model. We make use of the theory of Lie systems to provide a (nonlinear) superposition rule for the general solution of SISt model. To conclude, we present a quantum version of the same model, giving rise to a parametric family of SISt systems. We provide a superposition rule for the general solution of the quantum extension as well. [ABSTRACT FROM AUTHOR]

We deal with Euler‐type half‐linear second‐order differential equations, and our intention is to derive conditions in order their non‐trivial solutions are non‐oscillatory. This paper connects to the article P. Hasil, J. Šišoláková, M. Veselý: Averaging technique and oscillation criterion for linear and half‐linear equations, Appl. Math. Lett. 92 (2019), 62–69, where the corresponding oscillatory counterpart is studied and an oscillation criterion is established. The used effective technique for this investigation is the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This paper is completed by two corollaries and an example concerning the linear case when we obtain new results as well. [ABSTRACT FROM AUTHOR]

From the perspective of large‐scale system, a complex dynamical network can be regarded as the interconnected system with the node subsystem and link subsystem, which implies that the node subsystem and the link subsystem are the two main bodies of dynamical behaviors of network. Therefore, the whole stability of network is influenced by not only the dynamics of node subsystem but also the dynamics of link subsystem. According to the above view, the wholly asymptotical stability (WAS) is defined in this paper for the complex dynamical network with the model of differential equations. The WAS is used to describe the node subsystem achieves asymptotical stability when link subsystem achieves also the asymptotic stability in Lyapunov sense. For the WAS of the complex dynamical network, the corresponding criteria are derived by checking whether certain matrices are Hurwitz. The analysis results show that even if the isolated nodes are not asymptotically stable in Lyapunov sense, employing the dynamics of link can also force the node subsystem to achieve the asymptotical stability. Finally, the simulation examples show the validity of methods in this paper. [ABSTRACT FROM AUTHOR]

This paper introduces advances in solving space‐time conformable nonlinear partial differential equations (PDEs) and exact wave solutions for Oskolkov equations. To arrive at these advances, nonlinear PDEs with space and time conformable partial derivatives are reduced to differential equations with conformable derivatives by using new modifications of the exponential rational function method (ERFM). These modifications are efficious approaches for obtaining exact analytical solutions. An important result of this work is that it yields new exact wave solutions for space‐time conformable pseudo‐parabolic type equations. The graphical representations of the obtained solutions are shown to confirm the accuracy and efficiency of the suggested method. [ABSTRACT FROM AUTHOR]

Uncertain fractional‐order differential equations driven by the Liu process are of significance to depict the heredity and memory features of uncertain dynamical systems. This paper primarily analyses the reliability of the uncertain fractional‐order dynamic system with a state constraint. First, consider the possibility that the real dynamical system is actually limited; the state constraint is absorbed to the ordinary uncertain fractional‐order dynamical system. The concept of reliability of uncertain system is presented innovatively, which are ulteriorly formulated through the existing first‐hitting time theorem. Second, based on the proposed reliability and under a given sufficient condition, a novel uncertain fractional‐order dynamic system with a state constraint is modeled mathematically; corresponding minimum operation ability of the uncertain system is also given. Lastly, the uncertain fractional‐order dynamic system with a state constraint is applied to different physical and financial dynamical models. Analytic expressions of reliability indexes are derived to demonstrate the reasonableness of our model. Meanwhile, expected time response and American barrier option prices are calculated by using the predictor–corrector scheme. The sensitivity analysis is also presented for the numerical examples. [ABSTRACT FROM AUTHOR]

We study the process of two‐phase flow in thin porous media domains of Brinkman type. This is generally described by a model of coupled, mixed‐type differential equations of fluids' saturation and pressure. To reduce the model complexity, different approaches that utilize the thin geometry of the domain have been suggested. We focus on a reduced model that is formulated as a single nonlocal evolution equation of saturation. It is derived by applying standard asymptotic analysis to the dimensionless coupled model; however, a rigid mathematical derivation is still lacking. In this paper, we prove that the reduced model is the analytical limit of the coupled two‐phase flow model as the geometrical parameter of domain's width–length ratio tends to zero. Precisely, we prove the convergence of weak solutions for the coupled model to a weak solution for the reduced model as the geometrical parameter approaches zero. [ABSTRACT FROM AUTHOR]