Hybrid Differential Equations Research Articles

Study of a class of fractional-order evolution hybrid differential equations using a modified Mittag-Leffler-type derivative

This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In addition, two kinds of Ulam–Hyers (U–H) stability results are deduced for the mentioned problem. A pertinent example is given to verify the results.

Existence Theory and Stability Analysis to a Class of Hybrid Differential Equations using Confirmable Fractal Fractional Derivative

This research work is related to study a class of hybrid differential equations (HDEs) using conformablefractal-fractional derivative (CFFD). Fixed point theory, in particular Krasnoselskii’s fixed point theoremis implemented on the said problem to establish the conditions for which there exists at least one solution.Also, stability results of Ulam-Hyers (U-H) and U-H Rassias types are derived. At the end of the paper, weadded examples for the purpose of justification and strengthen of our derived results.

Initial Value Problem for Hybrid Improved Conformable Fractional Differential Equations with Retardation and Anticipation

This manuscript is devoted to proving some results concerning the existence of solutions for a class of initial value problem for nonlinear implicit fractional Hybrid differential equations and improved conformable fractional derivative. The result is based on a fixed point theorem due to Dhage. Further, an example is provided for the justification of our main result.

Nonlocal boundary value problems for functional hybrid differential equations involving generalized w-Caputo fractional operator

In this manuscript , we establish the existence and uniqueness of solutions for boundary value problems of nonlinear hybrid fractional differential equations involving generalized $\omega-$Caputo fractional derivatives. The proofs are based on Krasnoselskii fixed point theorem and some basic concept of $\omega-$Caputo fractional analysis. As application, an nontrivial example is given in the last part of this paper to illustrate our theoretical results.

Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags

Abstract This paper deals with some qualitative properties of solutions to nonlinear neutral hybrid differential equations connected to ψ-Caputo fractional derivative with time-varying lags. First, we demonstrate the problem possesses a mild solution uniquely where the source function may have temporal singularities. Second, in some cases, we indicate that the problem possesses a unique mild solution under some weaker conditions than the previous one. Third, we also obtain a result on a global mild solution for the problem. Finally, the results are further enriched by studying a new type of Ulam–Hyers stability for the main equation. The main results are obtained by applying the nice inequality, first proposed and proven in this paper. Some befit examples are given to justify the applicability of the main results.

On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations

Abstract Recently, fractals and fractional calculus have received much attention from researchers of various fields of science and engineering. Because the said area has been found applicable in modeling various real-world processes and phenomena. Hybrid differential equations (HDEs) play significant roles in mathematical modeling of various processes because the aforesaid equations incorporate different dynamical systems as specific cases. For instance, it is possible to model and describe non-homogeneous physical phenomena on using the said equations. Therefore, this research work is concerned with studying a class of nonlinear hybrid fractal–fractional differential equations. We develop the existence result for the qualitative study using a hybrid fixed point theorem. For the mentioned goal, a fixed point theory for the product of two operators is applied to deduce appropriate conditions for the existence of exactly one solution. Additionally, the stability result based on Ulam–Hyers is also deduced. The said stability results play an important role in numerical investigations. In addition, a numerical method based on Euler procedure is utilized to approximate the solution of the proposed problems. Various computational test problems are given to demonstrate the results. Also, using various fractal–fractional order values, several graphical presentations are given for the examples. The concerned analysis will help in investigating many real-world problems modeled using HDEs with fractal–fractional orders in the near future.

AN ALGORITHMIC APPROACH TO LOCAL SOLUTION OF THE NONLINEAR HIGHER ORDER ORDINARY HYBRID DIFFERENTIAL EQUATIONS

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature.

USING LEARNING ABILITIES OF COMPUTATION IN COUNTING

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature.

Analysis on existence of system of coupled multifractional nonlinear hybrid differential equations with coupled boundary conditions

<abstract><p>This article dealt with a class of coupled hybrid fractional differential system. It consisted of a mixed type of Caputo and Hilfer fractional derivatives with respect to two different kernel functions, $ \psi_{_1} $ and $ \psi_{_2} $, respectively, in addition to coupled boundary conditions. The existence of the solution of the system was investigated using the Dhage fixed point theorem. Finally, an illustration was presented to validate our findings.</p></abstract>

Analysis of a hybrid fractional coupled system of differential equations in $ n $-dimensional space with linear perturbation and nonlinear boundary conditions

<abstract><p>In this paper, we investigated $ n $-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in $ 3 $-dimensional space at the end of the paper.</p></abstract>

A Study on Some Conformable Fractional Implicit Hybrid Differential Equations with Delay

This paper deals with some existence results for a class of conformable implicit fractional differential Hybrid equations with delay. The results are based on some suitable fixed point theorems. In the last section, we provide different examples to illustrate our obtained results.

Dhage iteration method for an algorithmic approach to local solution of the nonlinear second order ordinary hybrid differential equations

Dhage iteration method for an algorithmic approach to local solution of the nonlinear second order ordinary hybrid differential equations

On New Solutions of Fuzzy Hybrid Differential Equations by Novel Approaches

The goal of this paper is to find the best of two sixth-order methods, namely, RK-Huta and RK–Butcher methods for solving the fuzzy hybrid systems. We state a necessary definition and theorem in terms of consistency for convergence, and finally, we compare the obtained numerical results of two different methods with analytical solution using two different numerical examples. In addition to that, we generalize the solutions obtained by RK-6 Huta and RK-6 Butcher methods (same order different stage methods) for both the problems we handled. We are proposing these two methods in order to reduce the error in accuracy and to establish these two methods are better than any other existing numerical methods. The best of two sixth-order methods are found by the error analysis study for both the problems. Also, we show whether the change in number of stages of same order methods affects the accuracy of the approximation or not.

A Hybrid Differential Equations Model for the Dynamics of Single and Double Strand Breaks of Cancer Cells Treated by Radiotherapy

According to the Target Theory, the tumor population is divided into multiple different subpopulations, called targets, based on the diverse effects of ionizing radiation on human cells. Radiation particles can cause single or double-strand break(s). As such, cells are divided into three subpopulations, namely cells with no DNA fragmentation, cells with DNA single-strand breaks, and cells with DNA double-strand breaks. This work introduces a hybrid differential equation model, with coefficients described by random variables representing transition rates between targets. The model is utilized to simulate the dynamics of targets and describes the cell damage heterogeneity and the repair mechanism between two consecutive dose fractions. Therefore, a new definition of tumor lifespan based on population size is achieved. Stability and bifurcation analysis are performed. Finally, the probability of target inactivity after radiation and the probability of target re-activation following the repair mechanism are evaluated with respect to the tumor lifespan.

Existence and uniqueness of solutions to nonlinear impulsive hybrid differential equations with linear and nonlinear perturbations

Existence and uniqueness of solutions to nonlinear impulsive hybrid differential equations with linear and nonlinear perturbations

Theoretical and Computational Aspects of Fractional Hybrid Differential Equations

Theoretical and Computational Aspects of Fractional Hybrid Differential Equations

STUDY OF COUPLED SYSTEM OF FRACTIONAL HYBRID DIFFERENTIAL EQUATIONS VIA THE PRIOR ESTIMATE METHOD

The purpose of this paper is to establish some sufficient conditions needed for the existence and uniqueness of solutions to the coupled system of fractional hybrid differential equations (FHDEs). We make use of the prior estimate method to support the main results. An example is also provided to justify our main results.

Existence of an infinite system of fractional hybrid differential equations in a tempered sequence space

In this article, the existence of the solution of infinite system of fractional hybrid differential equations with initial conditions by reducing them to integral equations in the space \(C(I, m(\theta )^{\beta })\) is discussed based on Darbo fixed point theorem. In the end, a suitable example to strengthen the article is included.

Fuzzy fractional hybrid differential equations

This article is related to present and solve the theory of fractional hybrid differential equations with fuzzy initial values involving the fuzzy Riemann-Liouville fractional differential operators of order $0 < q < 1$. For the concerned presentation, we study the existence and uniqueness of a fuzzy solution are brought in detail basing on the concept of generalized division of fuzzy numbers. We have developed and investigated a fuzzy solution of a fuzzy fractional hybrid differential equation. At the end we have given an example is provided to illustrate the theory.

A Coupled System of Caputo–Hadamard Fractional Hybrid Differential Equations with Three-Point Boundary Conditions

This article presents a study of the existence and uniqueness of solutions for a system of hybrid fractional differential equations involving fractional derivatives of the Caputo-Hadamard type with three-point hybrid boundary conditions. In addition to this, the “Hyres–Ulam” stability of the solutions for this type of equation is verified, and finally a numerical example was presented to support our theoretical results.