Narrow Results

In this paper, we employed a novel technique with the help of the Daftardar-Gejji and Jafari method (DGJM) to solve the generalized time-fractional Cattaneo model (GTFCM) via Caputo derivative with order $1<\alpha <2$. The existence and uniqueness of solutions are examined using Banach’s fixed-point theorem, and stability is evaluated under Ulam–Hyers conditions. We compared our results with the exact solution and existing techniques, demonstrating our approach’s efficiency.

The analysis of nonlinear models and their complex geometrical shapes with globalized piecewise fractional operators for past heredity characteristics serve as the novelty of and motivation for this research. We explore the dynamics of a financial model in the aforementioned framework, using the well known and extensively applied numerical technique of the fractional piecewise Adams–Bashforth method. The effect of the employed parameters on price index, interest rate, investment demand, variation, and expenditure per investment from the saving amount along with the elasticity of the commercial markets demand are investigated. Various fractional orders are simulated in the sense of the Caputo and Atangana–Baleanu Caputo derivatives in the first and second subintervals, respectively. Moreover, the piecewise existence and uniqueness of the solution for the proposed model are established. The concept of Ulam–Hyers stability is applied to the obtained solution. This study will be helpful for beginners to complex geometrical and crossover dynamical problems in various physical and biological sciences.

This study investigates the chaotic behavior and stability of a nonlinear fractional-order financial system utilizing the Caputo fractional derivative. Initially, we formulate the nonlinear fractional-order financial model and establish the problem framework. Next, we prove the existence and uniqueness of solutions by applying the Banach and Schauder fixed-point theorems to the proposed system. Additionally, we analyze the Ulam–Hyers stability and discuss other significant findings related to the system’s stability. To simulate the proposed model, we develop numerical schemes based on fractional calculus, employing Lagrange polynomial interpolation. Finally, we present the numerical simulations to validate the theoretical results, highlighting the significant impact of fractional-order derivatives on the system’s behavior.

Fish farming, a potential activity for the production of protein with high nutritional value, has emerged as a substantial source of income in the modern world. The degradation of natural materials uses oxygen to reduce them below the life-support stage. A huge area with an excessive amount of water swamp for oxygen exchange makes for the ideal ecosphere for fish farming, yet the mussel population may grow swiftly in a really modest water swapping farm producing a lot of feed. In this study, we examine the dynamics and chaotic behavior of a fish farm model containing populations of nutrients and mussels with different kernels insight of fractal-fractional operator. We examined the positively invariant area as well as demonstrate positive, bounded solutions of the model. We also show the equilibrium states for the occurrence and extinction of species. We prove the existence and uniqueness of positive solutions through fixed-point theorems and verified the Ulam–Hyers stability for proposed model. To study the impact of the fractional operator through computational simulations, results are generated employing a two-step Lagrange polynomial in the generalized version for the various kernels, which include the power-law and Mittag–Leffler kernel. We develop some numerical simulations to test our theoretical conclusions with chaotic behavior at different fractional-order values.

In this study, we establish a sophisticated compartmental Caputo fractional-order Susceptible-Vaccinated-Exposed-Infected-Hospitalized-Recovered Pathogens mathematical model for analyzing the dynamical behavior of COVID-19 transmission in the perspective of COVID-19 pandemic situation of Bangladesh considering the data up to 2nd dose vaccination. The frame of Caputo fractional-order derivative has been used to fractionalize the model’s equations. A total of 10 distinct compartments have been used to represent the dynamics of human and novel coronavirus. The Next-generation matrix approach has been used for finding the basic reproduction number of considered model. For establishing the existence and uniqueness of the model’s solutions, here we apply Banach fixed point theorem. Routh–Hurwitz, Ulam–Hyers and generalized Ulam–Hyers stability criteria have been applied for stability analysis. We execute a sensitivity analysis to check the comparative prominence of the model’s parameters. For the numerical simulation, data have been accumulated from the COVID-19 dashboard of Directorate general of health services of Bangladesh. The Fractional Adams–Bashforth–Moulton method along with MATLAB code fde12.m has also been applied to perform the numerical simulation. Furthermore, here we use some Maple codes for finding the model’s equilibrium points, basic reproduction number and sensitivity indices. Finally, we include a discussion on the influence of new rate of infections through environment and interaction with infected persons as well as the effect of 1st and 2nd dose vaccinations.

In this paper, new model on novel coronavirus disease (COVID-19) with four compartments including susceptible, exposed, infected, and recovered class with fractal-fractional derivative is proposed. Here, Banach and Leray–Schauder alternative type theorems are used to establish some appropriate conditions for the existence and uniqueness of the solution. Also, stability is needed in respect of the numerical solution. Therefore, Ulam–Hyers stability using nonlinear functional analysis is used for the proposed model. Moreover, the numerical simulation using the technique of fundamental theorem of fractional calculus and the two-step Lagrange polynomial known as fractional Adams-Bashforth (AB) method is proposed. The obtained results are tested on real data of COVID-19 outbreak in Malaysia from 25 January till 10 May 2020. The numerical simulation of the proposed model has performed in terms of graphs for different fractional-order $q$ and fractal dimensions $p$ via number of considered days of disease spread in Malaysia. Since COVID-19 transmits rapidly, perhaps, the clear understanding of transmission dynamics of COVID-19 is important for countries to implement suitable strategies and restrictions such as Movement Control Order (MCO) by the Malaysian government, against the disease spread. The simulated results of the presented model demonstrate that movement control order has a great impact on the transmission dynamics of disease outbreak in Malaysia. It can be concluded that by adopting precautionary measures as restrictions on individual movement the transmission of the disease in society is reduced. In addition, for such type of dynamical study, fractal-fractional calculus tools may be used as powerful tools to understand and predict the global dynamics of the mentioned disease in other countries as well.

In this paper, a new set of differential and integral operators has recently been proposed by Abdon et al. by merging the fractional derivative and the fractal derivative, taking into account nonlocality, memory and fractal effects. These operators have demonstrated the complex behavior of many physical, which generally does not predict in ordinary operators or sometimes in fractional operators also. In this paper, we investigate the proposed model by replacing the classic derivative by fractal–fractional derivatives in which fractional derivative is taken in Atangana–Baleanu Caputo sense to study the complex behavior due to nonlocality, memory and fractal effects. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model posseses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By means of nonlinear functional analysis, we prove that the proposed model is Ulam–Hyers stable under the new fractal–fractional derivative. We establish the numerical results of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we study the chaos behavior of the proposed model via simulation at 2D and 3D phase. We show the effect of fractal dimension on integer and fractional order through simulations.

This paper discussed a new operator known as a fractal-fractional (FF), considered in the Caputo sense. We have investigated the fractional mathematical model of Tuberculosis (TB) disease under FF Caputo derivative. We have provided the existence and uniqueness for the appropriate system by using Banach and Leray–Schauder theorem and for the stability of the model, we have used the Ulam–Hyers approach. Applying the methods of basic theorems of FF calculus (FFC) and the iterative numerical techniques of fractional Adams–Bashforth method for approximate solution. For the simulation of the model, we have considered different values for fractional order $\alpha$ and fractal dimension $\beta$ and compared the results with integer order for real data. The FFC technique is applied as a beneficial technique to know about the real-world problem and also to control the whole world situation of the aforesaid pandemic in the different continents and territories of the world. This new operator, FF in the form of Caputo derivative, gives better results than ordinary integer order. Several results have been discussed by taking the different fractal dimensions and arbitrary order.

In this paper, we solve a nonlinear fractional-order model for analyzing the dynamical behavior of vector-borne diseases within the frame of Caputo-fractional derivative. The proposed mathematical model advances the existing integer-order model on transmission and cure of vector-borne diseases. The existence and uniqueness of the solutions of the fractional-order model are proved using the Banach contraction principle. We investigate the local asymptomatic stability for the obtained disease-free equilibrium point and global stability for the proposed model in the sense of Ulam–Hyers stability criteria, respectively. Besides that, we obtain a numerical solution for the projected model using the Corrector-Predictor algorithm. Finally, to illustrate the obtained theoretical results, we perform numerical simulations for different values of fractional-order derivative and make a comparison with the results of the integer-order derivative.

This paper is devoted to the study of the dynamical behavior of the tritrophic interaction amongst plants, herbivores and carnivores mathematical model, expressed by three nonlinear ordinary differential equations under fractal–fractional derivative in the Caputo sense. We use fixed point theory to ensure that one solution exists to the proposed model. In addition, Hyers–Ulam’s stability analysis is studied by using theorem of functional analysis. For the numerical solution, we apply the fractional Adams–Bashforth iterative technique. For arbitrary fractional order and fractal dimensions, we study the dynamical and chaotic behavior of the obtained results for the considered model. Using Matlab 16, the system is then solved to get the required numerical solution for the proposed system. From the numerical simulations, we observed that the decay in fractional order dynamics of the system is stabilized when the amplitude of the oscillations becomes smaller.

In this study, we proposed a novel approach for modeling the dynamics of a three-agent financial bubble using the fractal-fractional (FF) derivative of the Caputo sense. This new concept was developed to deal with the complex geometry of any dynamical system, and it utilizes both the fractional derivative for the order and the fractal term for the order of the independent variables. The model was investigated using the conformable order derivative of the Caputo operator, with a focus on the fractal dimension and fractional order. The existence and uniqueness of the solution were tested using a FF global derivative, and the approximate root of the system was calculated using the numerically iterative technique of the Newton polynomial. To verify the accuracy of the approximate root scheme, we applied the power singular law with two fractional parameters in the developed numerical technique. The curve representation of the system was also verified by applying the data for the fractals and different conformable orders. Additionally, we tested the sensitivities of the fractional parameters and dynamical system parameters by varying the parameter values. This allowed us to gain a better understanding of how changes in these parameters affect the system’s behavior and stability. As a result, this study presents an innovative and effective approach for modeling the dynamics of financial bubbles using the FF derivative of the Caputo sense. The results of this research contribute to the ongoing efforts to develop more accurate and comprehensive models of complex systems in economics and finance.

In this paper, Pachpatte’s inequality is employed to discuss the Ulam–Hyers stabilities for Volterra integrodifferential equations and Volterra delay integrodifferential equations in Banach spaces on both finite and infinite intervals. Examples are given to show the applicability of our obtained results.

This paper deals with the Ulam–Hyers stability of the following additive-quadratic mixed type functional equation:

This study first focuses on defining a new fashion of multi-cubic mappings, and then represent such mapping as an single equation, as the multi-cubic functional equation. Additionally, this work also discusses the Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability of multi-cubic equations by applying the so-called direct (Hyers) method in the setting of 2-Banach spaces.

The fuzzy variable fractional differential equations (FVFDEs) play a very important role in mathematical modeling of COVID-19. The scientists are studying and developing several aspects of these COVID-19 models. The existence and uniqueness of the solution, stability analysis are the most common and important study aspects. There is no study in the literature to establish the existence, uniqueness, and UH stability for fuzzy variable fractional (FVF) order COVID-19 models. Due to high demand of this study, we investigate results for the existence, uniqueness, and UH stability for the considered COVID-19 model based on FVFDEs using a fixed point theory approach with the singular operator. Additionally, discuss the maximal/minimal solution for the FVFDE of the COVID-19 model.

This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration. Moreover, with the help of fixed point theory, we study the existence and uniqueness conditions for the positive solution and prove some new results. Also, obtain the Ulam–Hyers stabilities for the proposed problem. Two generalized examples are considered to show the method’s applicability and compared with other existing numerical methods. The present method performs exceptionally well in terms of efficiency and simplicity. Further, we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.