Staff

Abstract:
Precipitation nowcasting in real-time is a challenging task that demands accurate and current data from multiple sources. Despite various approaches proposed by researchers to address this challenge, models such as the interaction-based dual attention LSTM (IDA-LSTM) face limitations, particularly in radar echo extrapolation. These limitations include higher computational costs and resource requirements. Moreover, the fixed kernel size across layers in these models restricts their ability to extract global features, focusing more on local representations. To address these issues, this study introduces an enhanced convolutional long short-term 2D (ConvLSTM2D) based architecture for precipitation nowcasting. The proposed approach includes time-distributed layers that enable parallel Conv2D operations on each image input, enabling effective analysis of spatial patterns. Following this, ConvLSTM2D is applied to capture spatiotemporal features, which improves the model’s forecasting skills and computational efficacy. The performance evaluation employs a real-world weather dataset benchmarked against established techniques, with metrics including the Heidke skill score (HSS), critical success index (CSI), mean absolute error (MAE), and structural similarity index (SSIM). ConvLSTM2D demonstrates superior performance, achieving an HSS of 0.5493, a CSI of 0.5035, and an SSIM of 0.3847. Notably, a lower MAE of 11.16 further indicates the model’s precision in predicting precipitation.

Keywords:
precipitation nowcasting, radar echo, spatiotemporal dynamics

Abstract:
Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution.

Keywords:
Approximate solution, Caputo derivative, Reductive perturbation, Homotopy perturbation method

Abstract:
Mathematical models are powerful tools to study various real-world problems from different perspectives. This branch has been given much more popularity over the last several decades. Various mathematical models corresponding to different diseases have been studied so far. Keeping these details in mind, the present manuscript is devoted to present a detailed mathematical analysis of the Cutaneous Leishmaniasis disease model. Some basic properties of the model are studied including positivity, the existence of equilibrium points, and reproductive number. The existence and uniqueness of the solution for the model under consideration are also investigated. Local and global stability analyses of equilibrium points are also studied. For the required results, we use the Lyapunov function method and the third additive compound matrix technique based on the Metzler procedure. Sensitivity analysis is also investigated by using some tools from the numerical-functional analysis. A numerical analysis of the proposed model is performed by using a nonstandard finite difference scheme. Moreover, for the justification of our results, we give some graphical presentation of the model for each class in the model. Also, we present some graphical presentations related to the sensitivity analysis along with the tables for its various indices.

Keywords:
Leishmaniasis Disease model, Local and global stability, Sensitivity Analysis, Numerical analysis, Non standard finite difference method

Abstract:
In this research work, we present a mathematical analysis of a fractional sixth-order laser model of a resonant which is homogeneously extended three-level optically pumped. We use Caputo fractional order derivative in the proposed model. Our analysis includes an investigation of various chaotic behaviors under fractional order derivative and qualitative theory of the existence of the solution to the proposed model. For our required analysis of qualitative type, we use formal analysis tools. Further, numerical simulations are performed with a clustering method based on the K–Means algorithm and Adams Bashforth scheme. With the help of the aforesaid scheme, we present different chaotic behavior corresponding to various values of fractional order. Finally, we give a comparison of the CPU time of the proposed method with that of the RK4 method.

Keywords:
Qualitative theory, Chaotic behavior, K–Means algorithm, A clustering method, Adams Bashforth Scheme

Abstract:
Short memory and long memory terms are excellently explained using the concept of piecewise fractional order derivatives. In this research work, we investigate dynamical systems addressing COVID-19 under piecewise equations with fractional order derivative (FOD). Here, we study the sensitivity of the proposed model by using some tools from the nonlinear analysis. Additionally, we develop a numerical scheme to simulate the model against various fractional orders by using Matlab 2016. All the results are presented graphically.

Keywords:
Nonlinear dynamical system, Crossover behavior, Mathematical biology, Sensitivity analysis

Abstract:
In this work, we modified a dynamical system that addresses COVID-19 infection under a fractal-fractional-order derivative. The model investigates the psychological effects of the disease on humans. We establish global and local stability results for the model under the aforementioned derivative. Additionally, we compute the fundamental reproduction number, which helps predict the transmission of the disease in the community. Using the Carlos Castillo-Chavez method, we derive some adequate results about the bifurcation analysis of the proposed model. We also investigate sensitivity analysis to the given model using the criteria of Chitnis and his co-authors. Furthermore, we formulate the characterization of optimal control strategies by utilizing Pontryagin’s maximum principle. We simulate the model for different fractal-fractional orders subject to various parameter values using Adam Bashforth’s numerical method. All numerical findings are presented graphically.

Keywords:
dynamical system, fractal-fractional-order derivative, Pontryagin’s maximum principle, bifurcation analysis, sensitivity analysis, control strategies

Abstract:
This article is devoted to investigate a mathematical model consisting on susceptible, exposed, infected, quarantined, vaccinated, and recovered compartments of COVID-19. The concerned model describes the transmission mechanism of the disease dynamics with therapeutic measures of vaccination of susceptible people along with the cure of the infected population. In the said study, we use the fractal-fractional order derivative to understand the dynamics of all compartments of the proposed model in more detail. Therefore, the first model is formulated. Then, two equilibrium points disease-free (DF) and endemic are computed. Furthermore, the basic threshold number is also derived. Some sufficient conditions for global asymptotical stability are also established. By using the next-generation matrix method, local stability analysis is developed. We also attempt the sensitivity analysis of the parameters of the proposed model. Finally, for the numerical simulations, the Adams–Bashforth method is used. Using some available data, the results are displayed graphically using various fractal-fractional orders to understand the mechanism of the dynamics. In addition, we compare our numerical simulation with real data in the case of reported infected cases.

Abstract:
The second-hand smoke is a phenomenon that needs to be investigated, and its effects on the health of the people are to be examined. To analyze such an issue, the mathematical models are the best tools that help us to study the dynamical behaviors of this phenomenon. For this purpose, in the present paper, we consider a three-compartmental fractal-fractional mathematical model of a specific population of smokers or people that are exposed to second-hand smoke. By assuming some conditions on ϕ-ψ-contractions and compact operators, we prove some theorems in relation to the existence of solutions. The Banach principle for the usual contractions is used for proving the uniqueness of solutions. Next, by some notions of functional analysis, two types of Ulam-Hyers stability for the fractal-fractional second-hand smoker model are established. Moreover, we have a steady-state analysis and obtain equilibrium points and basic reproduction number R0. Then, we investigate the sensitivity of the fractal-fractional system with respect to each parameter. For numerical simulation, the Adams-Bashforth (AB) method is used to derive numerical schemes for plotting and simulating the approximate solutions. Finally, the obtained solutions are tested with real data and different values of fractal dimensions and fractional orders.

Abstract:
This study proposes a novel mathematical model of COVID-19 and its qualitative properties. Asymptotic behavior of the proposed model with local and global stability analysis is investigated by considering the Lyapunov function. The mentioned model is globally stable around the disease-endemic equilibrium point conditionally. For a better understanding of the disease propagation with vaccination in the population, we split the population into five compartments: susceptible, exposed, infected, vaccinated, and recovered based on the fundamental Kermack-McKendrick model. He's homotopy perturbation technique is used for the semi-analytical solution of the suggested model. For the sake of justification, we present the numerical simulation with graphical results.

Keywords:
Local asymptotic stability, global asymptotic stability, Routh-Hurwitz criterion, COVID-19, infectious disease modeling

Abstract:
In our observation, we have used an easy and reliable approach of the reduction perturbation method to obtain the solution of the ion temperature gradient mode driven linear and nonlinear structures of relatively small amplitude. One can use that methodology in the more complex environment of the plasma and can obtain a straightforward approach toward his studies. We have studied different parameter impacts on the linear and nonlinear modes of the ITG by using data from tokamak plasma. Hence, our study is related to the tokamak plasma and one that can apply to the nonlinear electrostatic study of stiller and interstellar regimes where such types of plasma environment occur.

Keywords:
Ion temperature gradient, soliton, shock, electron-ion plasma, reduction perturbation method, linear and non-linear structures

Abstract:
In the present work, we investigated the transmission dynamics of fractional order SARS-CoV-2 mathematical model with the help of Susceptible , Exposed , Infected , Quarantine , and Recovered . The aims of this work is to investigate the stability and optimal control of the concerned mathematical model for both local and global stability by third additive compound matrix approach and we also obtained threshold value by the next generation approach. The author’s visualized the desired results graphically. We also control each of the population of underlying model with control variables by optimal control strategies with Pontryagin’s maximum Principle and obtained the desired numerical results by using the homotopy perturbation method. The proposed model is locally asymptotically unstable, while stable globally asymptotically on endemic equilibrium. We also explored the results graphically in numerical section for better understanding of transmission dynamics.

Keywords:
Basic reproduction number, Stability analysis, Third additive compound matrix, Homotopy perturbation method, Next generation matrix, Fractional optimal control

Abstract:
In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

Abstract:
In the work, author’s presents a very significant and important issues related to the health of mankind’s. Which is extremely important to realize the complex dynamic of inflected disease. With the help of Caputo fractional derivative, We capture the epidemiological system for the transmission of Novel Coronavirus-19 Infectious Disease (nCOVID-19). We constructed the model in four compartments susceptible, exposed, infected and recovered. We obtained the conditions for existence and Ulam’s type stability for proposed system by using the tools of non-linear analysis. The author’s thoroughly discussed the local and global asymptotical stabilities of underling model upon the disease free, endemic equilibrium and reproductive number. We used the techniques of Laplace Adomian decomposition method for the approximate solution of consider system. Furthermore, author’s interpret the dynamics of proposed system graphically via Mathematica, from which we observed that disease can be either controlled to a large extent or eliminate, if transmission rate is reduced and increase the rate of treatment.

Keywords:
Fractional Derivatives, Fixed point theory, Ulams type Stabilities, Mathematical modeling, Approximate Solutions, Laplace-Adomian decomposition method

Abstract:
In this paper, we consider a mathematical model of Rabies disease which is an infectious disease. The model we are considering is a system of nonlinear ordinary differential equations and it is difficult to find an exact solution. He’s Homotopy perturbation method is employed to compute an approximation to the solution of the system of nonlinear ordinary differential equations. The findings obtained by HPM are compared with a nonstandard finite difference (NSFD) and Runge-Kutta fourth order (RK4) methods. Some plots are presented to show the reliability and simplicity of the method.

Keywords:
Mathematical model, infectious, nonlinear ordinary, homotopy, equations