Search

Goal: To accurately detect infections in Diabetic Foot Ulcers (DFUs) using photographs taken at the Point of Care (POC). Achieving high performance is critical for preventing complications and amputations, as well as minimizing unnecessary emergency department visits and referrals. Methods: This paper proposes the Guided Conditional Diffusion Classifier (ConDiff). This novel deep-learning framework combines guided image synthesis with a denoising diffusion model and distance-based classification. The process involves (1) generating guided conditional synthetic images by injecting Gaussian noise to a guide (input) image, followed by denoising the noise-perturbed image through a reverse diffusion process, conditioned on infection status and (2) classifying infections based on the minimum Euclidean distance between synthesized images and the original guide image in embedding space. Results: ConDiff demonstrated superior performance with an average accuracy of 81% that outperformed state-of-the-art (SOTA) models by at least 3%. It also achieved the highest sensitivity of 85.4%, which is crucial in clinical domains while significantly improving specificity to 74.4%, surpassing the best SOTA model. Conclusions: ConDiff not only improves the diagnosis of DFU infections but also pioneers the use of generative discriminative models for detailed medical image analysis, offering a promising approach for improving patient outcomes.

Abstract In this paper, we investigate the fractional hybrid integro-differential equations with Dirichlet boundary conditions. We first prove the existence of a unique solution for the equation using a fixed point technique. Our main goal is to obtain the best approximation using optimal controllers. After studying the stability, we present the reproducing kernel Hilbert space numerical method to obtain approximate solutions to the equation. We finally conclude with numerical results.

Abstract To make better decisions on approximation, we may need to increase reliable and useful information on different aspects of approximation. To enhance information about the quality and certainty of approximating the solution of an Apollonius-type quadratic functional equation, we need to measure both the quality and the certainty of the approximation and the maximum errors. To measure the quality of it, we use fuzzy sets, and to achieve its certainty, we use the probability distribution function. To formulate the above problem, we apply the concept of Z-numbers and introduce a special matrix of the form diag ( A , B , C ) $\mathrm{diag}(A, B, C)$ (named the generalized Z-number) where A is a fuzzy time-stamped set, B is the probability distribution function, and C is a degree of reliability of A that is described as a value of A ∗ B $A\ast B$ . Using generalized Z-numbers, we define a novel control function to investigate H–U–R stability to approximate the solution of an Apollonius-type quadratic functional equation with quality and certainty of the approximation.

To make a decision to select a suitable approximation for the solution of a functional inequality, we need reliable information. Two useful information ideas are quality and certainty, and the measure of quality and certainty approximation of the solution of a functional inequality helps us to find the optimum approximation. To measure quality and certainty, we used the idea of the Z-number (Z-N) and we introduced the generalized Z-N (GZ-N) as a diagonal matrix of the form $ diag(X, Y, X\ast Y) $, where $ X $ is a fuzzy set time-stamped, $ Y $ is the probability distribution function and the third part is the fuzzy-random trace of the first and the second subjects. This kind of diagonal matrix allowed us to define a new model of control functions to stabilize our problem. Using stability analysis, we obtained the most suitable approximation for functional inequalities.

The main issue we are studying in this paper is that of aggregation maps, which refers to the process of combining various input values into a single output. We apply aggregated special maps on Mittag-Leffler-type functions in one parameter to get diverse approximation errors for fractional-order systems in Hilfer sense using an optimal method. Indeed, making use of various well-known special functions that are initially chosen, we establish a new class of matrix-valued fuzzy controllers to evaluate maximal stability and minimal error. An example is given to illustrate the numerical results by charts and tables.

Abstract Introducing a new generalized multivariate Mittag-Leffler function which is a generalization of the multivariate Mittag-Leffler function, we derive a sufficient condition for the uniqueness of solutions to a brand new boundary value problem of the fractional nonlinear partial integro-differential equation using Banach’s fixed point theorem and Babenko’s technique. This has many potential applications since uniqueness is an important topic in many scientific areas, and the method used clearly opens directions for studying other types of equations and corresponding initial or boundary value problems. In addition, we use Python which is a high-level programming language efficiently dealing with the summation of multi-indices to compute approximate values of the generalized Mittag-Leffler function (it seems impossible to do so by any existing integral representation of the Mittag-Leffler function), and provide an example showing applications of key results derived.

In this paper, we consider a differential equation system and present a new method based on Clique polynomials (CP-M) to obtain numerical solutions of this system. The system of differential equations is a mathematical model of a new virus called Corona, which causes an infectious disease called COVID-19. By solving this system of equations, we check the transmissibility of the Coronavirus by the CPs method. In particular we turn the system of differential equations into an algebraic system to obtain solutions. Finally, we compare the numerical results obtained by the CPs method with the numerical results of other methods.

Abstract In this paper, we apply some special functions to introduce a new class of control functions that help us define the concept of multi-stability. Further, we investigate the multi-stability of homomorphisms in C ∗ $C^{*}$ -algebras and Lie C ∗ $C^{*}$ -algebras, multi-stability of derivations in C ∗ $C^{*}$ -algebras, and Lie C ∗ $C^{*}$ -algebras for the following random operator equation via fixed point methods: μ f ( ð , x + y 2 ) + μ f ( ð , x − y 2 ) = f ( ð , μ x ) . $$ \mu f \biggl(\eth , \frac{x+y}{2} \biggr) + \mu f \biggl(\eth , \frac{x-y}{2} \biggr) = f(\eth , \mu x) . $$ In particular, for μ = 1 $\mu = 1$ , the above equation turns out to be Jensen’s random operator equation.

Abstract The purpose of this work is to investigate the necessary conditions for the existence and uniqueness of solutions, and to introduce a new idea of α-confluent-hyper-geometric stability of an impulsive fractional differential equation with ξ-Hilfer fractional derivative. We use the Diaz–Margolis fixed point theorem to achieve this and illustrate the result with an example.

In this paper, we apply some special functions (Mittag-Leffler, Gauss Hypergeometric, Bessel-Maitland and Fox H functions) to investigate a class of matrix-valued random control functions. Using this kind of control function helps us to introduce the concept of multi-stability and get an approximation of non homogenous vector-valued fractional differential equations via the fixed point method. Further, some Ulam-Hyers stability results for the mentioned fractional differential equations in different cases are obtained. Also, we determine some sufficient conditions for the stability of non homogenous vector-valued fractional differential equations.

The present paper considers a fractional-order smoke epidemic model. We apply fuzzy systems and probability theory to make the best decision on the stability of the smoking epidemic model by using a new class of controllers powered by special functions to effectively generalize Ulam-type stability problems. Evaluation of optimal controllability and maximal stability is the new issue. This different concept of stability not only covers the old concepts but also investigates the optimization of the problem. Finally, we apply a new optimal method for the governing model with the Atangana–Baleanu–Caputo fractional derivative to obtain stability results in Banach spaces.

We apply known special functions from the literature (and these include the Fox H–function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the G–function, the Fox–Wright function and the Meijer G–function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory.

The quantum Hall effect under the influence of gravity and inertia is studied in a unified way. We make use of an algebraic approach, as opposed to an analytic approach. We examine how both the integer and the fractional quantum Hall effects behave under a combined influence of gravity and inertia using a unified Hamiltonian. For that purpose, we first re-derive, using the purely algebraic method, the energy spectrum of charged particles moving in a plane perpendicular to a constant and uniform magnetic field either (i) under the influence of a nonlinear gravitational potential or (ii) under the influence of a constant rotation. The general Hamiltonian for describing the combined effect of gravity, rotation and inertia on the electrons of a Hall sample is then built and the eigenstates are obtained. The electrons mutual Coulomb interaction that gives rise to the familiar fractional quantum Hall effect is also discussed within such a combination.

Abstract In this paper, we consider the nonhomogeneous fractional delay oscillation equation with order κ and investigate the existence of a unique solution in matrix-valued fuzzy Banach spaces for this equation using the alternative fixed point theorem. In a fuzzy environment, we introduce a class of the matrix-valued fuzzy Wright controller to investigate the Hyers–Ulam–Wright stability for the NH-FD-O equation with order κ. Finally, an illustrative example to demonstrate the application of the main theorem is also considered.

In this study, we investigate the concept of antiderivations in Banach algebras and study multi-super-stability of antiderivations in Banach algebras, associated with functional inequalities.

This paper focuses on obtaining traveling wave solutions of the Fornberg–Whitham model derived from Gilson–Pickering equations, which describe the prorogation of waves in crystal lattice theory and plasma physics by some analytical techniques, i.e., the exp-function method (EFM), the multi-exp function method (MEFM) and the multi hyperbolic tangent method (MHTM). We analyze and compare them to show that MEFM is the optimum method.

In this paper, we apply two different methods, namely, the G′G-expansion method and the G′G2-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation.

In this paper, using the fractional integral with respect to the Ψ function and the Ψ-Hilfer fractional derivative, we consider the Volterra fractional equations. Considering the Gauss Hypergeometric function as a control function, we introduce the concept of the Hyers-Ulam-Rassias-Kummer stability of this fractional equations and study existence, uniqueness, and an approximation for two classes of fractional Volterra integro-differential and fractional Volterra integral. We apply the Cădariu-Radu method derived from the Diaz-Margolis alternative fixed point theorem. After proving each of the main theorems, we provide an applied example of each of the results obtained.

In this article, we used direct algebraic method (DAM) and sine-Gordon expansion method (SGEM), to find the analytical solutions of conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE) and finally, we present numerical results in tables and charts.

Abstract We stabilize pseudostochastic ( G 1 , G 2 ) $(\mathcal{G}_{1},\mathcal{G}_{2})$ -random operator inequality using a class of stochastic matrix control functions in matrix Menger Banach algebras. We get an approximation for stochastic ( G 1 , G 2 ) $(\mathcal{G}_{1},\mathcal{G}_{2})$ -random operator inequality by means of both direct and fixed point methods. As an application, we apply both stochastic Mittag-Leffler and H $\mathbb{H}$ -fox control functions to get a better approximation in a random operator inequality.

We apply CRM based on an alternative FPT to investigate the approximation of a Δ-Hilfer FDE. In comparison to the Picard method, we show that the CRM has a better error estimate and economic solution.

In this paper, we consider the nonlinear fractional Laplacian pseudoparabolic equation (NFLPPE). We mainly focus on the convergence of mild solutions with respect to the order of fractional Laplacian. By using many techniques, we obtain the result that the mild solution will converge when the fractional order of the Laplacian tends to 1−. The proof of convergent result relies on sharp techniques of evaluating the exponential terms, the Sobolev embeddings, and weakly singular Gronwall inequalities.

In a random operator inequality, by the fuzzy controllers, we stable an approximately additive odd random operator and find an estimation for such random operators and we solve Hyers-Ulam-Rassias stability problem for a random operator inequality.

We introduce a matrix-valued fractional delay differential system in diverse cases and present Fox type stability results with applications of aggregated special functions. In addition we present an example showing the numerical solutions based on the second type Kudryashov method. Finally, via the method of variation of constants, and some properties of the parametric Mittag–Leffler matrices, we obtain both symmetric random and symmetric fuzzy finite-time stability results for the governing fractional delay model. A numerical example is considered to illustrate applicability of the study.

We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the stability of functional equations. We also present a small list of aggregation functions on the classical, well-known special functions to investigate the best approximation error estimates using a different concept of perturbation stability.

Abstract The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule.

Abstract In this article, we introduce a new concept of Hausdorff distance through generalized modular metric on nonempty compact subsets and study some topological properties of it. This concept with contraction theory and the iterated function system (IFS) helps us to define a generalized modular fractal space.

Abstract We propose a method for constructing a generalized fuzzy Hausdorff distance on the set of the nonempty compact subsets of a given generalized fuzzy metric space in the sence of George–Veeramni and Tian–Ha–Tian. Next, we define the generalized fuzzy fractal spaces. Morever, we obtain a fixed point theorem of a class of generalized fuzzy contractions and present an application in integranl equation.

In this paper, we apply the well-known aggregation mappings on Mittag-Leffler-type functions to investigating new approximation error estimates of a W-Hilfer fractional differential equation, by a different concept of Ulam-type stability in both bounded and unbounded domains.

This paper studies the uniqueness of the bounded solution to a new Cauchy problem of the fractional nonlinear partial integro-differential equation based on the multivariate Mittag–Leffler function as well as Banach’s contractive principle in a complete function space. Applying Babenko’s approach, we convert the fractional nonlinear equation with variable coefficients to an implicit integral equation, which is a powerful method of studying the uniqueness of solutions. Furthermore, we construct algorithms for finding analytic and approximate solutions using Adomian’s decomposition method and recurrence relation with the order convergence analysis. Finally, an illustrative example is presented to demonstrate constructions for the numerical solution using MATHEMATICA.

Stability is the most relevant property of dynamical systems. The stability of stochastic differential equations is a challenging and still open problem. In this article, using a fuzzy Mittag–Leffler function, we introduce a new fuzzy controller function to stabilize the stochastic differential equation (SDE) ν′(γ,μ)=Fγ,μ,ν(γ,μ). By adopting the fixed point technique, we are able to prove the fuzzy Mittag–Leffler–Hyers–Ulam–Rassias stability of the SDE.

Abstract We introduce a mathematical model, namely, ∗-fuzzy measure model for COVID-19 disease and consider some properties of ∗-fuzzy measure such as Lebesque–Radon–Nikodym theorem.

Abstract We consider fuzzy sets and generalized triangular norms on positive elements of order commutative C ∗ $C^{*}$ -algebras to study the concept of C ∗ $C^{*}$ -algebra valued normed algebras with uncertainty. Using n-expansively super-homogeneous and ( n , k ) $(n,k)$ -contractively sub-homogeneous control functions, we make stochastic ( Θ , ϒ , Ξ ) $(\Theta,\Upsilon,\Xi )$ -derivations stable and get a better estimated error. We present some numerical examples of control functions and approximations to illustrate the applicability of the main results.

Abstract In this article, first, we present an example of fuzzy normed space by means of the Mittag-Leffler function. Next, we extend the concept of fuzzy normed space to matrix valued fuzzy normed space and also we introduce a class of matrix valued fuzzy control functions to stabilize a nonlinear ϕ-Hadamard fractional Volterra integro-differential equation. In this sense, we investigate the Ulam–Hyers–Rassias stability for this kind of fractional equations in matrix valued fuzzy Banach space. Finally, as an application, we investigate the Ulam–Hyers–Rassias stability using matrix valued fuzzy control function obtained through the Mittag-Leffler function.

Abstract In this article, we introduce a class of stochastic matrix control functions to stabilize a nonlinear fractional Volterra integro-differential equation with Ψ-Hilfer fractional derivative. Next, using the fixed-point method, we study the Ulam–Hyers and Ulam–Hyers–Rassias stability of the nonlinear fractional Volterra integro-differential equation in matrix MB-space.

Abstract We introduce a class of stochastic matrix control functions and apply them to stabilize pseudo stochastic κ-random operator inequalities in matrix MB-algebras. We obtain an approximation for stochastic κ-random operator inequalities and calculate the maximum error of the estimate.

In this paper, we consider a generalized triangular norm on matrix and introduce Menger Banach matrix valued spaces. Next, we apply a class of random matrix control functions to investigate the existence, uniqueness and HUR stability of a class of fractional Volterra integral equations. We use the Radu-Mihe method, to get an approximation for the mentioned fractional Volterra integral equation.

Introduction/purpose: The aim of the paper is to establish some coupled fixed point results in C*-algebra-valued b-metric spaces. Moreover, the obtained results are used to define the sufficient conditions for the existence of the solutions of some classes of integral equations. Methods: The method of coupled fixed points gives the sufficient conditions for the existence of the solution of some classes of integral equations. Results: New results were obtained on coupled fixed points in C * -algebra-valued b-metric space. Conclusion: The obtained results represent a contribution in the fixed point theory and open new possibilities of application in the theory of differential and integral equations.

Abstract In this paper, we consider C ∗ $C^{*}$ -algebra valued fuzzy normed spaces. We study the random integral equation ( 1 2 c ) ∫ x − c d x + c d u ( γ , τ , d 0 ) d τ = u ( γ , x , d ) $(\frac{1}{2c})\int _{x-cd}^{x+cd}u(\gamma ,\tau ,d_{0})\,d\tau =u( \gamma ,x,d)$ which is related to the stochastic wave equation. In addition, using a C ∗ $C^{*}$ -algebra valued fuzzy controller function, we consider its C ∗ $C^{*}$ -algebra valued fuzzy Hyers–Ulam stability.

Abstract We consider a nonlinear Cauchy problem involving the Ψ-Hilfer stochastic fractional derivative with uncertainty, and we give a stability result. Using fixed point theory, we are able to provide a fuzzy Ulam–Hyers–Rassias stability for the considered nonlinear stochastic fractional differential equations.

Abstract By a stochastic controller, we make stable the pseudo stochastic Lie bracket (derivation, derivation) in complex MB-algebras. Next, we get an approximation by a stochastic Lie bracket (derivation, derivation) and calculate the maximum error of the estimate.

In this article, we apply a new class of fuzzy control functions to approximate a Cauchy additive mapping in fuzzy Banach space (FBS). Further, considering the unital FBS (UFBS), we will investigate the isomorphisms defined in this space. By introducing several specific functions and choosing the optimal control function from among these functions, we evaluate the Cauchy–Optimal stability (C–O-stability) for all defined mappings.

Cortical neurons receive mixed information from the collective spiking activities of primary sensory neurons in response to a sensory stimulus. A recent study demonstrated an abrupt increase or decrease in stimulus intensity and the stimulus intensity itself can be respectively represented by the synchronous and asynchronous spikes of S1 neurons in rats. This evidence capitalized on the ability of an ensemble of homogeneous neurons to multiplex, a coding strategy that was referred to as synchrony-division multiplexing (SDM). Although neural multiplexing can be conceived by distinct functions of individual neurons in a heterogeneous neural ensemble, the extent to which nearly identical neurons in a homogeneous neural ensemble encode multiple features of a mixed stimulus remains unknown. Here, we present a computational framework to provide a system-level understanding on how an ensemble of homogeneous neurons enable SDM. First, we simulate SDM with an ensemble of homogeneous conductance-based model neurons receiving a mixed stimulus comprising slow and fast features. Using feature-estimation techniques, we show that both features of the stimulus can be inferred from the generated spikes. Second, we utilize linear nonlinear (LNL) cascade models and calculate temporal filters and static nonlinearities of differentially synchronized spikes. We demonstrate that these filters and nonlinearities are distinct for synchronous and asynchronous spikes. Finally, we develop an augmented LNL cascade model as an encoding model for the SDM by combining individual LNLs calculated for each type of spike. The augmented LNL model reveals that a homogeneous neural ensemble model can perform two different functions, namely, temporal- and rate-coding, simultaneously.

We apply special functions and use the concept of the aggregation function to introduce a new class of fuzzy control functions, and based on this, we obtain the best approximation for the stochastic bi-homomorphisms and stochastic bi-derivations in FB-algebras and FC-⋄-algebras of matrix type associated with the bi-additive random operator inequality.

We apply Mittag–Leffler-type functions to introduce a class of matrix-valued fuzzy controllers which help us to propose the notion of multi-stability (MS) and to obtain fuzzy approximate solutions of matrix-valued fractional differential equations in fuzzy spaces. The concept of multi stability allows us to obtain different approximations depending on the different special functions that are initially chosen. Additionally, using various properties of a function of Mittag–Leffler type, we study the Ulam–Hyers stability (UHS) of the models.

In this paper, we consider a conformable fractional differential equation with a constant coefficient and obtain an approximation for this equation using the Radu–Mihet method, which is derived from the alternative fixed- point theorem. Considering the matrix-valued fuzzy k-normed spaces and matrix-valued fuzzy H-Fox function as a control function, we investigate the existence of a unique solution and Hyers–Ulam-H-Fox stability for this equation. Finally, by providing numerical examples, we show the application of the obtained results.

We consider a class of nonautonomous cellular neural networks (CNNs) with mixed delays, to study the solutions of these systems which are type pseudo almost periodicity. Using general measure theory and the Mittag-Leffler function, we obtain the existence of unique solutions for cellular neural equations and investigate the Mittag-Leffler stability and attractiveness of pseudo almost periodic functions. We also present numerical examples to illustrate the application of our results.

Using the trial equation method (TEM) and modified trial equation method (MTEM), firstly, we find the analytical solutions of the conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE), and finally, we present numerical results in tables and charts.

In this paper, we consider the nonhomogeneous fractional delay oscillation equation with order σ and introduce a class of control functions, i.e., Wright functions. Next, we apply the Cădariu-Radu method to prove the existence of a unique solution and Hyers-Ulam-Rassias-Wright stability of the fractional delay oscillation equation. At the end of the article, by an example, we show the application of the obtained results.

The main goal of the paper is to approximate two types of inverse problems for conformable heat equation (or called parabolic equation with conformable operator); as follows, we considered two cases: the right hand side of equation such that Fx,t and Fx,t=φtfx. Up to now, there are very few surveys working on the results of regularization in Lp spaces. Our paper is the first work to investigate the inverse problem for conformable parabolic equations in such spaces. For the inverse source problem and the backward problem, use the Fourier truncation method to approximate the problem. The error between the regularized solution and the exact solution is obtained in Lp under some suitable assumptions on the Cauchy data.