Four Different Ulam-Type Stability for Implicit Second-Order Fractional Integro-Differential Equation with M-Point Boundary Conditions

In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the β –Ulam stability, β –Hyers–Ulam stability and β –Hyers–Ulam–Rassias stability for the said system on a compact interval and then extended it to an unbounded interval. We use Grönwall type inequality and evolution family as a basic tool for our results. We present an example to demonstrate the application of the main result.

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

In this work, we derive novel theoretical results concerning well-posedness and Ulam–Hyers stability. Specifically, we investigate the well-posedness of Caputo–Katugampola fractional stochastic delay integro-differential equations. Additionally, we develop a generalized Grönwall inequality and apply it to prove Ulam–Hyers stability in Lp space. Our findings generalize existing results for fractional derivatives and space, as we formulate them in the Caputo–Katugampola fractional derivative and Lp space. To support our theoretical results, we present an illustrative example.

In this paper, we derive general solution of new n-dimensional quintic and sextic functional equations and investigate the Hyers–Ulam stability, Hyers–Ulam–Rassias stability and generalized Hyers–Ulam–Rassias stability for quintic and sextic functional equations in Felbin type fuzzy normed linear spaces. Also, we give the counter examples for the Hyers–Ulam–Rassias stability of quintic and sextic functional equations for some cases.

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

Hyers-Ulam stability and existence of solutions for weighted Caputo-Fabrizio fractional differential equations

In this paper, we prove the existence and uniqueness of solution for random fractional integro-differential equation. The existence of at least mean square continuous solution for this problem is discussed. Using the fixed point theorem, Ulam–Hyers stability and Ulam–Hyers–Rassias stability of random fractional integro-differential equation are studied. Finally, we give an example to illustrate our results.

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.

In the field of functional equations and their solutions, Ulam's stability is an essential concept. This theory examines whether the function approximating a certain functional equation is close to the function that exactly satisfies it. A broader extension of the stability concept is generalized Hyers–Ulam stability. Classifying, analysing and solving functional equations across multiple spaces are made easier by using this. In this investigation, we examine the generalized Hyers–Ulam stability of generalized cubic- and quartic-type functional equations of the form: m { f ( m ϑ + μ ) + f ( m ϑ − μ ) } + f ( ϑ + m μ ) + f ( ϑ − m μ ) = 2 m 2 { f ( ϑ + μ ) + f ( ϑ − μ ) } + 2 ( m 4 − 2 m 2 + 1 ) f ( ϑ ) and f ( m ϑ + μ ) + f ( m ϑ − μ ) + f ( ϑ + m μ ) + f ( ϑ − m μ ) = 2 m 2 + { f ( ϑ + μ ) + f ( ϑ − μ ) } + 2 ( m 4 − 2 m 2 + 1 ) { f ( ϑ ) + f ( μ ) } , in setting of non-Archimedean (n-A) normed spaces by using distinguished Hyers direct method. Also, we provide a graphical representation of an approximate solution and how it differs from an exact solution for both equations. Furthermore, we present counterexamples that demonstrate the failure case of stability.

ABSTRACTThe article deals with a second kind of hybrid noninstantaneous impulsive fractional differential equation with ‐Laplacian operator involving Caputo–Katugampola fractional derivatives. The rudimentary concept of this article is to establish the existence, uniqueness, Hyers–Ulam stability, and Hyers–Ulam–Rassias stability. The fixed point approach is the basis for establishing the results of solution to the considered problem. An example is also given to highlight the main findings.

The study of Ulam stability for (functional, differential, difference, integral, integro-differential, and fractional differential) equations heavily relies on inequalities. In such scientific and engineering research, fixed point theorems (FPTs) are essential instruments. This article, on one hand, focused on the $ \beta $-Ulam-Hyers stability ($ \beta $-UHS) of non-instantaneous impulsive fractional integro-differential equations (N-IIFIDEs) involving the Caputo-Fabrizio fractional derivatives (C-FFDs) in a Banach space. On the other hand, we established the existence and uniqueness (E-UR) of solutions by employing the Banach contraction mapping principle (BCMP) and Krasnoselskii's fixed point theorem (KFPT). To validate the theoretical insights, a carefully crafted example was introduced. In this way, we generalized recent interesting results.

In this article, we consider a new class of fractional impulsive neutral stochastic functional integro-differential equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, fractional calculus, analytic α-resolvent operator and suitable fixed point theorems, we prove the existence of mild solutions and optimal mild solutions for these equations. Second, the existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. The results are obtained under weaker conditions in the sense of the fractional power arguments. Finally, an example is given for demonstration.

The purpose of this paper is to study the Hyers–Ulam stability and Generalized Hyers–Ulam stability of the general Linear Differential Equations of first order and second order with constant coefficients using Fourier Transform method. Moreover, the Hyers–Ulam stability constants of these differential equations are obtained. Some examples are given to illustrate the main results.

On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions