Laplace Adomian Decomposition Method Research Articles

Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

PurposeFractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.Design/methodology/approachThis paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.FindingsThis paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.Originality/valueThis paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

Analyzing Co-Infection Dynamics: A Mathematical Approach Using Fractional Order Modeling and Laplace-Adomian Decomposition

The co-infection of HIV and COVID-19 is a pressing health concern, carrying substantial potential consequences. This study focuses on the vital task of comprehending the dynamics of HIV-COVID-19 co-infection, a fundamental step in formulating efficacious control strategies and optimizing healthcare approaches. Here, we introduce an innovative mathematical model grounded in Caputo fractional order differential equations, specifically designed to encapsulate the intricate dynamics of co-infection. This model encompasses multiple critical facets: the transmission dynamics of both HIV and COVID-19, the host’s immune responses, and the influence of treatment interventions. Our approach embraces the complexity of these factors to offer an exhaustive portrayal of co-infection dynamics. To tackle the fractional order model, we employ the Laplace-Adomian decomposition method, a potent mathematical tool for approximating solutions in fractional order differential equations. Utilizing this technique, we simulate the intricate interactions between these variables, yielding profound insights into the propagation of co-infection. Notably, we identify pivotal contributors to its advancement. In addition, we conduct a meticulous analysis of the convergence properties inherent in the series solutions acquired through the Laplace-Adomian decomposition method. This examination assures the reliability and accuracy of our mathematical methodology in approximating solutions. Our findings hold significant implications for the formulation of effective control strategies. Policymakers, healthcare professionals, and public health authorities will benefit from this research as they endeavor to curtail the proliferation and impact of HIV-COVID-19 co-infection.

Mathematical modeling of chickenpox transmission using the Laplace Adomian Decomposition Method

In this work, mathematical modeling of a nonlinear differential equation was studied to investigate the effect of vaccination on the spread of chickenpox. The proof of existence and uniqueness of the positive solution and invariant region showed that the model is epidemiologically sound. We established the disease-free and endemic equilibrium states and carried out a stability analysis of the disease-free and endemic equilibrium states of the model to gain insight into the dynamics of the model. The rate of vaccination and precaution for the spread of chickenpox was a factor that influenced the basic reproductive number, which was calculated using the next-generation matrix approach. Forecasts made via numerical simulation using the Laplace Adomian Decomposition method highlight the temporal impact of vaccination on curbing the chicken pox trend.

The analytical analysis of fractional differential system via different operators and normalization functions

This article is related to the utilization of the Laplace Adomian Decomposition Method (LADM) to analyze the solution of a fractional order system under different operators and normalized functions. In this connection, the three new and updated operators i.e., Caputo, Caputo–Fabrizio, and Atangana–Baleano are used to define the fractional order derivative with the help of four different normalization functions. It is observed that, among the four normalized functions, the fourth kind of normalized function is the most suitable function for the above-mentioned operators. The LADM series solutions are obtained for both integer and fractional orders, implying that the proposed approach has a greater accuracy and convergence rate. Figures and Tables are constructed to verify the validity and efficiency of the suggested technique. LADM simulations confirmed, that the more iterations of LADM reduced the associated absolute error. Moreover, fractional order solutions are obtained which are convergent towards integer order solutions.

The analysis of a novel COVID-19 model with the fractional-order incorporating the impact of the vaccination campaign in Nigeria via the Laplace-Adomian Decomposition Method

This study underscores the crucial role of COVID-19 vaccinations in managing the pandemic, with a specific focus on Nigeria. Employing a fractional-order mathematical modeling approach, the research assesses vaccination efficacy, minimum effectiveness, and duration. The model’s numerical solution is derived through the Laplace Adomian Decomposition Method (LADM), utilizing rapidly converging infinite series. Simulation results illustrate the impact of COVID-19 transmission and vaccination rates. The study concludes that implementing a vaccination strategy in an integer order proves to be the most effective approach to controlling the spread of COVID-19. These findings have significant implications for researchers, policymakers, and healthcare workers. They emphasize the central role of fractional calculus in facilitating vaccine implementation in the ongoing battle against COVID-19. The study calls for global efforts to maximize vaccination implementation for the overall benefit of public health.

Modeling and dynamics of measles via fractional differential operator of singular and non-singular kernels

Young children frequently die from measles, which is a major global health concern. Despite being more prevalent in infants, pregnant women, and people with compromised immune systems, it can infect anyone. Novel fractional operators, the constant-proportional Caputo operator, and the constant-proportional Atangana–Baleanu operator are used to create a hybrid fractional order model that helps analyze the dynamic transmission of the measles virus. We assess the measles-free and endemic equilibrium, reproductive number, biological viability, boundedness, well-posedness, and positivity of the model. We apply the Banach contraction principle to verify the uniqueness of the system’s solutions. The proposed system is confirmed to be Ulam–Hyres stable by using fixed point theory results. The aforementioned operators are further analyzed, and the Laplace-Adomian decomposition method is used to numerically simulate the system of fractional differential equations. To support our findings, the outcomes are graphically displayed. The efficacy and memory impact of fractional operators are illustrated through comparisons. Based on fractional parameter values, the study determines important disease-control strategies and shows that vaccinations greatly reduce the spread of measles. By reducing the number of infected people, increasing vaccination coverage lowers the burden of disease on the general population.

Approximate Solution of the Fractional Order Mathematical Model on the Transmission Dynamics on The Co-Infection of COVID-19 and Monkeypox Using the Laplace-Adomian Decomposition Method

A fractional order compartmental model on the transmission dynamics of the co-infection of COVID-19 and Monkeypox is presented. The approximate solutions of the fractional order model are obtained using the Laplace-Adomian Decomposition method in the form of an infinite series which was shown to converge to the exact value. Using the MATLAB fmincon algorithm, we carried out a data fitting analysis using real life COVID-19 and Monkeypox data so as to obtain estimates for some of the key parameters used in the formulation of model. The results of our analysis showed that an increase in the effective treatment capacity in the human population will significantly reduce the burden of these diseases in the human population.

Approximate Solution of Fractional Order of Partial Differential Equations Using Laplace-Adomian Decomposition Method in MATLAB

This article presents the application of the Laplace-Adomian Decomposition Method (LADM) for solving partial differential equations (PDEs) in the context of heat conduction and wave propagation. The LADM combines Laplace transform and Adomian decomposition to approximate solutions to PDEs efficiently in MATLAB. The procedure involves transforming the PDE into simpler differential equations, which are then solved iteratively using the Adomian decomposition method. The advantages of LADM include simplicity, flexibility, and applicability to a wide range of PDEs. We demonstrate the effectiveness of LADM through numerical experiments solving the heat equation and wave equation using MATLAB. The results show good agreement with analytical solutions and highlight the efficiency and accuracy of LADM for solving PDEs.

Analysis of a mathematical model with hybrid fractional derivatives under different kernel for hearing loss due to mumps virus

ABSTRACT Genetics, birth defects, some viral diseases, persistent ear infections, medications, loud noises, and age can all cause hearing loss, a global health issue. The mumps are typically the cause of unilaterally acquired sensorineural hearing loss in children. A hybrid fractional operator model employing Constant Proportional Caputo (CPC), Constant Proportional Caputo Fabrizio (CPCF), and Constant Proportional Atangana-Baleanu Caputo (CPABC) in the Caputo sense is developed to investigate mumps induced hearing loss in children. We focused on the region that is positively invariant and well-posed in the suggested model. We assessed the endemic and disease free equilibrium, reproductive number, strength number, and parameter sensitivity, and examined the global stability of the proposed system. The existence, uniqueness, and Hilfer generalized proportional operators for the constant of proportionality of the system’s solutions are demonstrated using fixed-point theorems. We simulated a system of fractional differential equations analytically using the Laplace Adomian Decomposition Method (LADM). This study is important since it is the first to model mumps related hearing loss using a fractional order model based on hybrid fractional operators without a singular kernel. The fractional-order derivative retains the system memory effect, is nonlocal, and outperforms the integer-order derivative in this regard. This sort of study aids medical professionals in diagnosis, treatment planning, and decision-making for mumps induced hearing loss.

A caputo fractional-order model of tuberculosis incorporating enlightenment and therapy using the Laplace-Adomian decomposition method

ABSTRACT In this study, we used a deterministic model to investigate tuberculosis transmission dynamics while considering control measures. We analyzed four compartmental models representing susceptible, latent, infected, and recovered populations. To assess the potential spread of the disease, we calculated the basic reproduction number using the next-generation matrix after establishing a disease-free equilibrium. Additionally, we explored the endemic equilibrium, as well as local and global stabilities. To obtain numerical solutions, we employed the Laplace-Adomian decomposition method. For constructing the model, we utilized Maple 18 software and varied the order (0 < ψ < 1) to examine the impact of enlightenment and therapy. Graphical representations were used to analyze the effects of non-integer order Caputo derivative parameters on the susceptible, latent, infected, and recovered populations over time. Our detailed discussion highlights the crucial role of mathematical models based on Caputo fractional derivatives in effectively controlling and eradicating tuberculosis from society. These findings contribute valuable insights to inform strategies for combating the disease and promoting public health initiatives.

Investigation of Soret and Dufour Effects on Chemically Reacting Free Convective Fluid Flowing Over a Vertical Plate Along with Viscous Dissipation Using the Laplace Adomian Decomposition Method

This study comprehensively analyses heat and mass transfer phenomena in a chemically reacting free convective fluid flow along a vertically moving plate. The flow is influenced by thermo-diffusion, diffusion-thermo, and viscous dissipation effects. To simplify the analysis, scaling group analysis and appropriate similarity transformations are used to transform the governing equations into nonlinear ordinary differential equations. These equations are then solved using a combination of Laplace transform and the Adomian decomposition method. The study conducts a parametric investigation to explore the impact of various control parameters on the dimensionless velocity, temperature, and concentration profiles. The parameters considered include the Prandtl number, Schmidt number, Eckert number, chemical reaction parameter, Soret parameter, Dufour parameter, solutal Grashof number, and thermal Grashof number. These parameters are depicted graphically and analysed quantitatively. The results reveal that an increase in the Schmidt number leads to a decrease in velocity and concentration profiles while temperature varies monotonically. Elevating the Eckert number enhances velocity and temperature profiles, with a slight decrease in concentration profiles. A rise in the Prandtl number decreases the temperature profile, with minimal effects on velocity and concentration profiles. Increasing the solutal Grashof number decreases temperature and concentration profiles, whereas the thermal Grashof number is directly proportional to the velocity profile. An increase in the Dufour parameter boosts velocity and temperature profiles while reducing the concentration profile. The presence of the Soret parameter increases velocity and concentration profiles but decreases the temperature profile. This study aims to enhance comprehension of the complex interactions within flow characteristics, providing valuable insights into the fundamental mechanisms of such systems. It also highlights their potential applications in various engineering and industrial processes.

A note on Laplace Adomian decomposition method on a linear problem

In last two decades, Laplace Adomian Decomposition Method (LADM) is vastly used to solve non-linear (or even fractional order) differential equations. The method approaches the solution with the partial sums of function series. However, it is not easy to show that the limit of the function series is the exact solution of the problem. In this article, we consider a simple problem such as homogenous second-order linear ordinary differential equation with constant coefficients. We prove analytically that the LADM gives the right exact solution to the considered problem.

Designing a novel fractional order mathematical model for COVID-19 incorporating lockdown measures

This research focuses on the design of a novel fractional model for simulating the ongoing spread of the coronavirus (COVID-19). The model is composed of multiple categories named susceptible S(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(t)$$\\end{document}, infected I(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I(t)$$\\end{document}, treated T(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(t)$$\\end{document}, and recovered R(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(t)$$\\end{document} with the susceptible category further divided into two subcategories S1(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${S}_{1} (t)$$\\end{document} and S2(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${S}_{2} (t)$$\\end{document}. In light of the need for restrictive measures such as mandatory masks and social distancing to control the virus, the study of the dynamics and spread of the virus is an important topic. In addition, we investigate the positivity of the solution and its boundedness to ensure positive results. Furthermore, equilibrium points for the system are determined, and a stability analysis is conducted. Additionally, this study employs the analytical technique of the Laplace Adomian decomposition method (LADM) to simulate the different compartments of the model, taking into account various scenarios. The Laplace transform is used to convert the nonlinear resulting equations into an equivalent linear form, and the Adomian polynomials are utilized to treat the nonlinear terms. Solving this set of equations yields the solution for the state variables. To further assess the dynamics of the model, numerical simulations are conducted and compared with the results from LADM. Additionally, a comparison with real data from Italy is demonstrated, which shows a perfect agreement between the obtained data using the numerical and Laplace Adomian techniques. The graphical simulation is employed to investigate the effect of fractional-order terms, and an analysis of parameters is done to observe how quickly stabilization can be achieved with or without confinement rules. It is demonstrated that if no confinement rules are applied, it will take longer for stabilization after more people have been affected; however, if strict measures and a low contact rate are implemented, stabilization can be reached sooner.

LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations

Abstract Generally, fractional partial integro-differential equations (FPIDEs) play a vital role in modeling various complex phenomena. Because of the several applications of FPIDEs in applied sciences, mathematicians have taken a keen interest in developing and utilizing the various techniques for its solutions. In this context, the exact and analytical solutions are not very easy to investigate the solution of FPIDEs. In this article, a novel analytical approach that is known as the Laplace adomian decomposition method is implemented to calculate the solutions of FPIDEs. We obtain the approximate solution of the nonlinear FPIDEs. The results are discussed using graphs and tables. The graphs and tables have shown the greater accuracy of the suggested method compared to the extended cubic-B splice method. The accuracy of the suggested method is higher at all fractional orders of the derivatives. A sufficient degree of accuracy is achieved with fewer calculations with a simple procedure. The presented method requires no parametrization or discretization and, therefore, can be extended for the solutions of other nonlinear FPIDEs and their systems.

Stability analysis of fractional order breast cancer model in chemotherapy patients with cardiotoxicity by applying LADM

AbstractBreast cancer is the most common type of cancer in women. Chemotherapy is primarily used for patients with stage 2 to 4 breast cancer. Most chemotherapy drugs are effective at destroying rapidly growing and proliferating cancer cells. However, drugs also damage normal, rapidly growing cells, which can lead to serious side effects. Breast cancer treatment with chemotherapy can affect heart health. Side effects of chemotherapy on the heart are called cardiotoxicity. Therefore, we have constructed a mathematical model from the breast cancer patient population. In this article, we utilize the Caputo–Fabrizio fractional order derivative for mathematical modeling of the breast cancer stages in chemotherapy patients. The use of Caputo–Fabrizio fractional derivative provides a more valuable insight into the complexity of the breast cancer model. The stability of the fractional order model is also proven by the $\mathscr{P}$ P -stable approach of the fixed point theorem. Also, the numerical simulations are performed via Laplace Adomian decomposition method to establish the dependence of the breast cancer dynamics on the order of the fractional derivatives. Based on the geometric results in the figures, we can conclude that the magnitude of the fractional order has a considerable impact on the days, which the maximum or minimum of the system solutions are reached, with a shift in the time at which this happens as the fractional order decreases from 1. However, it is obvious that the solutions of Caputo–Fabrizio fractional model approach the relevant results of the classical integer order system, when the fractional order approaches to 1.

Forecasting and dynamical modeling of reversible enzymatic reactions with a hybrid proportional fractional derivative

Chemical kinetics is a branch of chemistry that investigates the rates of chemical reactions and has applications in cosmology, geology, and physiology. In this study, we develop a mathematical model for chemical reactions based on enzyme dynamics and kinetics, which is a two-step substrate–enzyme reversible reaction, applying chemical kinetics-based modeling of enzyme functions. The non-linear differential equations are transformed into fractional-order systems utilizing the constant proportional Caputo–Fabrizio (CPCF) and constant proportional Atangana–Baleanu–Caputo (CPABC) operators. The system of fractional differential equations is simulated using the Laplace–Adomian decomposition method at different fractional orders through simulations and numerical results. Both qualitative and quantitative analyses such as boundedness, positivity, unique solution, and feasible concentration for the proposed model with different hybrid operators are provided. The stability analysis of the proposed scheme is also verified using Picard’s stable condition through the fixed point theorem.

Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative

Abstract This study demonstrates the use of fractional calculus in the field of epidemiology, specifically in relation to dengue illness. Using noninteger order integrals and derivatives, a novel model is created to examine the impact of temperature on the transmission of the vector–host disease, dengue. A comprehensive strategy is proposed and illustrated, drawing inspiration from the first dengue epidemic recorded in 2009 in Cape Verde. The model utilizes a fractional-order derivative, which has recently acquired popularity for its adaptability in addressing a wide variety of applicable problems and exponential kernel. A fixed point method of Krasnoselskii and Banach is used to determine the main findings. The semi-analytical results are then investigated using iterative techniques such as Laplace-Adomian decomposition method. Computational models are utilized to support analytical experiments and enhance the credibility of the results. These models are useful for simulating and validating the effect of temperature on the complex dynamics of the vector–host interaction during dengue outbreaks. It is essential to note that the research draws on dengue outbreak studies conducted in various geographic regions, thereby providing a broader perspective and validating the findings generally. This study not only demonstrates a novel application of fractional calculus in epidemiology but also casts light on the complex relationship between temperature and the dynamics of dengue transmission. The obtained results serve as a foundation for enhancing our understanding of the complex interaction between environmental factors and infectious diseases, leading the way for enhanced prevention and control strategies to combat global dengue outbreaks.

A constant proportional caputo operator for modeling childhood disease epidemics

We propose a constant proportional-Caputo (CPC) operator for modeling childhood disease epidemics by treating positivity, boundedness, well-posedness, and biological viability and checking the reproductive and strength numbers for the biological system. For local and global stability analysis, we used the Volterra-type Lyapunov function with the first and second derivatives, providing symmetric results for fractional-order derivatives. We used different techniques to invert the proportional-Caputo (PC) and constant proportional-Caputo operators to evaluate the fractional integral operator. We also used our suggested model’s fractional differential equations to derive the eigenfunctions of the CPC operator. Modern mathematical control with the fractional operator has many applications, including the complex and crucial study of systems with symmetry. Symmetry analysis is a powerful tool that enables the user to construct numerical solutions to a given fractional differential equation fairly systematically. Therefore, we construct the comparative results with the Caputo derivative and constant proportional-Caputo operator to understand the dynamic behavior of the disease. We also present an additional analysis of the constant proportional-Caputo and Hilfer generalized proportional operators. Using the Laplace Adomian Decomposition Method, we numerically simulate a system of fractional differential equations through tables and graphs.

NON- INTEGER MATHEMATICAL MODEL OF RESPIRATORY SYNCYTIAL VIRUS

Respiratory syncytial virus (RSV), also known as a human respiratory syncytial virus (hRSV) and human orthopneumovirus, is a common, transmittable virus that roots respiratory tract diseases. It is a negative-sense, single-stranded Ribonucleic acid (RNA) virus. It gets its name from syncytia, which are huge cells that form when infected cells merge. In this paper fractional order model of the respiratory syncytial virus (RSV) virus will be developed. The Caputo fractional derivative operator of the order will be used to generate the model scheme of non-integer differential equations. To calculate an estimated solution of the system of nonlinear fractional differential equations, the Laplace-Adomian Decomposition Method was used. Infinite series was produced as solutions to fractional differential equations. The model's proposed series solution converges quickly to its precise value. The obtained results are compared to the standard case.

A numerical approach for an epidemic SIR model via Morgan-Voyce series

Abstract This study presents the problem of spreading non fatal disease in a population by using the Morgan-Voyce collocation method. The main aim of this paper is to find the exact solutions of the SIR model with vaccination. The problem may be modelled mathematically with a nonlinear system of ordinary differential equations. The presented method reduces the problem into a nonlinear algebraic system of equations by using unknown coefficient Morgan-Voyce polynomials and expanding approximate solutions. Morgan-Voyce polynomials are used. These unknown coefficients are calculated via the collocation method and matrix operation derivations. Two examples are given to show the feasibility of the method. To calculate the solutions, MATLAB R2021a is used. Additionally, comparing our method to the Homotopy perturbation method (HPM) and the Laplace Adomian decomposition method (LADM) proves the accuracy of the solution. The method studied can be seen as effective from these comparisons. So, it is essential to find solutions for the governing model. The study will contribute to literature since we also discuss the vaccination situation. The results of this study are valuable for controlling an epidemic.