Search Results (55)

In this paper, we obtain unique solution and stability results for coupled fractional differential equations with p-Laplacian operator and Riemann–Stieltjes integral conditions that expand and improve the works of some of the literature. In order to obtain the existence and uniqueness of solutions for coupled systems, several fixed point theorems for operators in ordered product spaces are given without requiring the existence conditions of upper–lower solutions or the compactness and continuity of operators. By applying the conclusions of the operator theorem studied, sufficient conditions for the unique solution of coupled fractional integro-differential equations and approximate iterative sequences for uniformly approximating unique solutions were obtained. In addition, the Hyers–Ulam stability of the coupled system is discussed. As applications, the corresponding results obtained are well demonstrated through some concrete examples. Full article

The main goal of this research is to study integro-fractional differential equations and simulate their dynamic behavior using ABC-fractional derivatives. We investigate the Hyers–Ulam stability of the proposed system and further expand the prerequisites for the existence and uniqueness of the solutions. The Schauder fixed-point theorem and the Banach contraction principle are employed to obtain the results. Finally, we present an example to demonstrate the practical application of our theoretical conclusions. Full article

This paper, by constructing a fractional epidemic model, analyzes the transmission dynamics of some infectious diseases under the effect of vaccination, which is one of the most effective and common control measures. In the model, considering that antibody formation by vaccination may not cause permanent immunity, it has been taken into account that the protection period provided by the vaccine may be finite, in addition to the fact that this period may change according to individuals. The model differs from other $SVIR$ models given in the literature in its progressive process with a distributed delay in the loss of the protective effect provided by the vaccine. To explain this process, the model was constructed by using a system of distributed delay nonlinear fractional integro-differential equations. Thus, the model aims to present a realistic approach to following the course of the disease. Additionally, an analysis was conducted regarding the minimum vaccination ratio of new members required for the elimination of the disease in the population by using the vaccine free basic reproduction number (${\mathcal{R}}_0^{vf}$). After providing examples for the selection of the distribution function, the variation of ${\mathcal{R}}_0$ was simulated for a specific selection of parameters in the model. Finally, the sensitivity indices of the parameters affecting ${\mathcal{R}}_0$ were calculated, and this situation is been visually supported. Full article

Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial fractional derivative is also defined, and its properties are discussed. The Laplace transforms (LTs) of the introduced fractional operators are evaluated. The Hyers–Ulam stability and the existence of a novel tempered fractional differential equation are examined. Moreover, a fractional integro-differential kinetic equation is formulated, and the LT is used to find its solution. A growth model and its graphical representation are established, highlighting the role of novel fractional operators in modeling complex dynamical systems. The developed mathematical framework offers valuable insights into solving a range of scenarios in mathematical physics. Full article

This paper investigates a class of multi-agent systems (MASs) governed by nonlinear fractional-order space-varying partial integro-differential equations (SVPIDEs), which incorporate both nonlinear state terms and integro terms. Firstly, a distributed adaptive control protocol is developed for leaderless fractional-order SVPIDE-based MASs, aiming to achieve consensus among all agents without a leader. Then, for leader-following fractional-order SVPIDE-based MASs, the protocol is extended to account for communication between the leader and follower agents, ensuring that the followers reach consensus with the leader. Finally, three examples are presented to illustrate the effectiveness of the proposed distributed adaptive control protocols. Full article

This work studies an integro-fractional differential equation (I-FrDE) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (I-DE) into a mixed integral equation (MIE) with an Able kernel in fractional time and a generalized symmetric singular kernel in position. Additionally, the authors first set conditions on the singular kernels, whether related to time or position, and then transform the integral equation into an integral operator. Secondly, the solution is unique, which is proven by means of fixed-point theorems. In combination with the solution rules, the convergence of the solution is studied, and the error equation resulting from the solution is a stable error-integral influencer equation. Next, to solve this MIE, the authors apply a special technique to separate the variables and produce an integral equation in position with coefficients, in the form of an integral operator in time. As the most effective technique for resolving singular integral equations, the Toeplitz matrix method (TMM) is utilized to convert the integral equation into an algebraic system for the purpose of solving the position problem. The existence of a solution to the linear algebraic system in Banach space is then demonstrated. Lastly, certain applications where the functions of the generalized symmetric kernel are cubic or exponential and it assumes the logarithmic, Cauchy, or Carleman form are discussed. In each case, Maple 18 is also used to compute the error estimate. Full article

In studying boundary value problems and coupled systems of fractional order in $(1,2]$, involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in $(1,2],$ involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach’s fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results. Full article

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, electrostatics, control theories, and the natural sciences. An effective approach solves the problem by reducing it to a pair of algebraically separated equations via a successful transformation. The proposed strategy uses first-order shifted Chebyshev polynomials and a projection method. Using the provided technique, the primary system is converted into a set of algebraic equations that can be solved effectively. Some theorems are proved and used to obtain the upper error bound for this method. Furthermore, various examples are provided to demonstrate the efficiency of the proposed algorithm when compared to existing approaches in the literature. Finally, the key conclusions are given. Full article

A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The convergence analysis of Bell wavelets approximation is discussed. The fractional integro-differential equations can be simplified to a system of algebraic equations by using a truncated Bell wavelets series and the fractional operational matrix. The proposed method’s efficacy is supported via various examples. Full article

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed. Full article

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results. Full article

This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order $1<\alpha <2$ with delay in Banach spaces. Firstly, we study the existence of the $PC$-mild solution for our objective system via some characteristic solution operators related to the Mainardi’s Wright function. Secondly, by using the spatial decomposition techniques and the range condition of control operator B, some new results of approximate controllability for the fractional delay system with impulsive effects are obtained. The results cover and extend some relevant outcomes in many related papers. The main tools utilized in this paper are the theory of cosine families, fixed-point strategy, and the Grönwall-Bellman inequality. At last, an example is given to demonstrate the effectiveness of our research results. Full article

Proposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference scheme. Also, the temporal derivative is discretized via a second-order numerical procedure. First, the spatial derivatives are approximated by a fourth-order operator to compute the second-order derivatives. This process produces a system of differential equations related to the time variable. Then, the Crank–Nicolson idea is utilized to achieve a full-discrete scheme. The kernel of the integral term is discretized by using the Lagrange polynomials to overcome its singularity. Subsequently, we prove the convergence and stability of the new difference scheme by utilizing the Rayleigh–Ritz theorem. Finally, some numerical examples in one-dimensional and two-dimensional cases are presented to verify the theoretical results. Full article

In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space $L_2\left[\mathsf{\Omega }\right]\times C\left[0,T\right], T<1$. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS). Full article

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space $C_E\left(E, J\right)$. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters $\lambda$ and time $t$ as well as how the absolute error in each example is calculated. Full article