Natural Decomposition Method Research Articles

Existence and Numerical Investigation of Monkey-Pox Mathematical Model by Natural Adomain Decomposition Method

In this paper, studied the mathematical model concerning the transmission of Monkey-Pox disease. A class viral disease that mostly occurs in west and central Africa, transmitted from animals into human is belonging to the Small-pox family known is Monkey-pox infections disease. According to the scientist the primary best of the proposed disease is still in doubt. The proposed model will be investigate for the purpose of both qualitative and numerical solutions. At the early stage of this study, investigate the existence of proposed model. In this connection, the authors developed the desired condition of existence and stability for consider model by using the tools of analysis. At the second phase of this research work,the author investigated the numerical solutions for the consider Monkey-pox transmission diseases model. For numerical investigation, the authors use the tool of well know semi-analytical techniques known as Natural Transform coupled with Adomain Decomposition Method. The consider techniques are powerful tools for of obtaining approximate solutions of differential equation or system of differential equations. The proposed techniques base on recursive scheme for solutions of system of differential equations. For the authenticity and accuracy of obtain solutions, the obtain solutions are visualized graphically to desired the dynamical behavior of desired results with the help of Mathematica. That show the proposed method is best tools for solution of differential equations.

Solution analysis of Solow Growth Model for financial practices and applications

Solow Growth Model (SGM) is an economic model that is exogenous in nature and examines the relationship between the output and input levels in an economy over a period of time. It projects long-term economic growth in relation to labour (population growth), savings rate, and technological development. However, traditional approaches to solving the Solow growth model may rely on complex mathematical techniques that might not give an accurate representation of real-world economic dynamics. Thus, this paper applies the Natural Decomposition Method (NDM) to the Solow growth financial model. The NDM is a numerical technique that combines the Natural Transform (NT) and the ADM-Adomian decomposition method. The NDM simplifies problem-solving by converting the original differential equations into algebraic equations as regards limitations associated with nonlinear models. From the results obtained by applying the NDM to the Solow growth financial model, researchers and policymakers can better understand the interplay between financial variables, such as savings rates, investment, and capital allocation, and their impact on economic growth dynamics, as a systematic approach to capturing the complex relationship between finance and economic development within the Solow framework is ensured. Further research and application of the NDM can contribute to advancing the knowledge of economic dynamics and support evidence-based decision-making in economic policy.

A novel Adomian natural decomposition method with convergence analysis of nonlinear time-fractional differential equations

ABSTRACT The objective of the current article is to use the Banach fixed point theorem to present proofs for the existence and uniqueness theorems for a nonlinear time-fractional differential equation, namely, the linear and nonlinear, homogeneous and nonhomogeneous time-fractional diffusion equations. In addition, we explore exact solutions to nonlinear time fractional partial differential equations using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). To demonstrate the effectiveness of the suggested scheme, three examples are provided along with graphs and numerical tables to support our work. The outcomes demonstrate how effortless and effective the ANDM is for addressing fractional partial differential equations in multi-dimensional spaces, some of which we examined in this study as special cases. AMS Classifications 34A08, 65P99, 49J15, 35R11, 26A33, 74G10.

Implementation of an effective transform technique for convergence analysis of the fractional enzyme kinetic and blood alcohol level models

In the current study, we solve the nonlinear time fractional‐order enzyme kinematic system and the linear time fractional blood alcohol level using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). On the basis of the Banach fixed point theorem, we offer proofs for the existence and uniqueness theorems as they apply to nonlinear ordinary differential equations. Using the current method, approximate analytical solutions of nonlinear time fractional‐order enzyme kinematic system and exact solutions to the linear time fractional‐order blood alcohol level problem have been obtained. The outcomes indicate that the ANDM is highly efficient and useful. The method outlined in this paper is intended to be applied in subsequent work to handle comparable nonlinear problems connected to fractional calculus.

An Efficient Model for Solving Fractional Order Partial Differential Equations

To solve fractional order differential equations, a hybrid technique named “the natural Adomian decomposition method (NADM) with Caputo fractional operator (CFO)” has been applied. The combination approach incorporates both the natural transform method (NTM) and the Adomian decomposition method (ADM). Two challenges are overcome to validate and demonstrate the efficacy of the current process. It is also shown that the results acquired using the suggested technique are extremely like those obtained using other strategies. For a range of science and engineering difficulties, the proposed solution has been shown to be efficient, dependable, and simple to implement.

Convergence analysis of the effects of antiviral drug treatment in the fractional differential model of HIV-1 infection of CD4+ T-cells

ABSTRACT The time fractional-order nonlinear model of HIV-1 infection of T-cells with and without the rate of virus-producing cells’ death is very important in mathematical modeling. This paper presents proofs of the existence and uniqueness theorems using the Banach fixed point theorem, along with new approximate analytical solutions to two nonlinear fractional Equationuations characterizing HIV-1 infection in CD 4 + T-cells. We use a combination of two methods, namely the natural transform method to deal with the linear terms and the Adomian decomposition method to deal with the nonlinear terms. There are two nonlinear numerical examples presented in this work, namely, we find approximate solutions to the time fractional-order nonlinear model of HIV-1 infection of T cells with and without the rate of virus-producing cells’ death using an effective method called the Adomian natural decomposition method (ANDM). The present approach, which has numerous applications in the science and engineering fields, is a great alternative to the many existing methods for solving systems of differential Equationuations. It also holds great promise for additional real-world applications. The Mathematica 12 package is used to carry out some of the nonlinear term calculations and the graphs presented in this work.

Reliable iterative techniques for solving the KS equation arising in fluid flow

In this study, we examine the Kuramoto-Sivashinsky equation which is a nonlinear model that describes several physical and chemical events arising in fluid flow. The approximate analytical solution for the fractional KS (FKS) problem is calculated using the Temimi-Ansari method (TAM) and the natural decomposition method (NDM). The projected procedure (NDM) combines the adomian decomposition method with the natural transform. Each technique can deal with nonlinear terms without making any assumptions. The methodologies under consideration provide � � -curves that display the convergence window of the power series solution that approaches the exact solution. We explore two distinct examples to confirm the efficiency and applicability of the proposed strategies. The acquired outcomes are compared numerically with the q-homotopy analysis transform method (q-HATM). The numerical investigation is carried out to validate the precision and dependability of the approaches under consideration. Additionally, the nature of the outcomes gained has been displayed in a different order. The obtained results show that the proposed techniques are highly efficient and simple to use to analyze the behavior of other nonlinear models. Keywords: Kuramoto-Sivashinsky equation, Laplace transform, natural decomposition method (NDM), Temimi-Ansari method (TAM)

Analytical solution to fractional differential equation arising in thermodynamics

The analysis of nonlinear events related to physical phenomena is a popular issue in the modern day. The essential purpose of this work is to discover a novel approximate solution to the fractional nonlinear Benjamin Bona Mahony Peregrine Burgers equation (BBMPB) utilizing the natural decomposition method (NDM) of fractional order. The suggested approach provides analytical solutions that are extremely near to the exact solution obviating the complexities associated with many other approaches. The expected issue’s uniqueness theorem and convergence analysis are explored using Banach’s fixed-point theory. The reliability and accuracy of the recommended method were tested using numerical simulations. The graphs and tables reflect the results. The comparison of the suggested scheme’s solution with the exact solutions demonstrates that the scheme is efficient, methodical, and extremely exact in tackling nonlinear complicated phenomena.

Important Theories and Applications Arise in Physics and Engineering using a New Technique: The Natural Decomposition Method

In the current research, we used the natural Adomian decomposition method (NADM), which is a unique technique, to find new exact solutions to partial and ordinary differential equations, such as the Sawada-Kotera equation, the extended fifth-order Korteweg-de Vries (efKdV) equation, and the Caudrey-Dodd-Gibbon equation. We also provided proofs for newly discovered theorems related to natural transformation. Our solutions to both linear and nonlinear differential equations are provided in series form, and the suggested method has high convergence. To handle differential equations, including nonlinear and linear differential equations, the proposed scheme in this work can be simply viewed as an alternative to those found in the literature.

Fractal dynamics and computational analysis of local fractional Poisson equations arising in electrostatics

In this paper, the local fractional natural decomposition method (LFNDM) is used for solving a local fractional Poisson equation. The local fractional Poisson equation plays a significant role in the study of a potential field due to a fixed electric charge or mass density distribution. Numerical examples with computer simulations are presented in this paper. The obtained results show that LFNDM is effective and convenient for application.

Novel Computations of the Time-Fractional Coupled Korteweg–de Vries Equations via Non-Singular Kernel Operators in Terms of the Natural Transform

In the present research, we establish an effective method for determining the time-fractional coupled Korteweg–de Vries (KdV) equation’s approximate solution employing the fractional derivatives of Caputo–Fabrizio and Atangana–Baleanu. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. Some theoretical physical features of quantum mechanics are also explained by the KdV model. Many investigations have been conducted on this precisely solvable model. Numerous academics have proposed new applications for the generation of acoustic waves in plasma from ions and crystal lattices. Adomian decomposition and natural transform decomposition techniques are combined in the natural decomposition method (NDM). We first apply the natural transform to examine the fractional order and obtain a recurrence relation. Second, we use the Adomian decomposition approach to the recurrence relation, and then, using successive iterations and the initial conditions, we can establish the series solution. We note that the proposed fractional model is highly accurate and valid when using this technique. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Two examples are given to illustrate how the technique performs. Tables and 3D graphs display the best current numerical and analytical results. The suggested method provides a series form solution, which makes it quite easy to understand the behavior of the fractional models.

The dynamics of fractional KdV type equations occurring in magneto-acoustic waves through non-singular kernel derivatives

To study magneto-acoustic waves in plasma, we will use a numerical method based on the Natural Transform Decomposition Method (NTDM) to find the approximative solutions of nonlinear fifth-order KdV equations. The method combines the familiar Natural transform (NT) with the standard Adomian decomposition method. The fractional derivatives considered are the Caputo–Fabrizio and the Atangana–Baleanu derivatives in the sense of Caputo derivatives. Adomian polynomials may be employed to tackle nonlinear terms. In this method, the solution is calculated as a convergent series, and it is demonstrated that the NTDM solutions converge to the exact solutions. A range of two- and three-dimensional figures have been used to illustrate the dynamic behavior of the derived solutions. The tables provide a visual representation of numerical data. The physical behavior of the derived solutions about fractional order is further demonstrated by several simulations. When addressing nonlinear wave equations in science and engineering, the NTDM offers a broad range of applications. Several examples are given to highlight the importance of this work and to demonstrate the simplicity and trustworthiness of the method.

Solutions of fuzzy advection-diffusion and heat equations by natural adomian decomposition method

In this article, we present an algorithm for computing analytical solutions of linear fuzzy advection-diffusion equations and one-dimensional fuzzy heat equations involving an external source. The fuzzy problems can be solved by using the natural transform and Adomian decomposition method. The results obtained through the natural Adomian decomposition method are calculated in a series form that converges rapidly to the exact solution. To enhance the practicality of our work, we provide examples to illustrate our findings.

Analytical and Approximate Solutions for Fractional Systems of Nonlinear Differential Equations

This paper studies the epidemic model of rational order via an efficient genius modern analytical approximate technique named Natural Transform Adomian Decomposition Method (NTADM). It is based on Caputo fractional derivative. To demonstrate the effectiveness of the present method, the results are displayed in graphs. Accordingly, the NTADM can be very easily applied to other nonlinear models.

Solutions of Time Fractional (1 + 3)-Dimensional Partial Differential Equations by the Natural Transform Decomposition Method (NTDM)

The current study employs the natural transform decomposition method (NTDM) to test fractional-order partial differential equations (FPDEs). The present technique is a mixture of the natural transform method and the Adomian decomposition method. For the purpose of checking the precis of our technique, some examples are offered, and the series solutions of these equations are introduced by using NTDM. The outcome shows that the suggested approach is very active and straightforward for obtaining a series solutions of FPDEs and is more accurate if we compare it with existing methods.

Approximate Solution of Painlevé Equation I by Natural Decomposition Method and Laplace Decomposition Method

Abstract The Painlevé equations and their solutions occur in some areas of theoretical physics, pure and applied mathematics. This paper applies natural decomposition method (NDM) and Laplace decomposition method (LDM) to solve the second-order Painlevé equation. These methods are based on the Adomain polynomial to find the non-linear term in the differential equation. The approximate solution of Painlevé equations is determined in the series form, and recursive relation is used to calculate the remaining components. The results are compared with the existing numerical solutions in the literature to demonstrate the efficiency and validity of the proposed methods. Using these methods, we can properly handle a class of non-linear partial differential equations (NLPDEs) simply. Novelty One of the key novelties of the Painlevé equations is their remarkable property of having only movable singularities, which means that their solutions do not have any singularities that are fixed in position. This property makes the Painlevé equations particularly useful in the study of non-linear systems, as it allows for the construction of exact solutions in certain cases. Another important feature of the Painlevé equations is their appearance in diverse fields such as statistical mechanics, random matrix theory and soliton theory. This has led to a wide range of applications, including the study of random processes, the dynamics of fluids and the behaviour of non-linear waves.

Theoretical analysis of new techniques applied to applications in fluid dynamics

ABSTRACT In this research study, we explore new proofs of theorems related to a novel technique by the name of the natural Adomian decomposition method (NADM), and then we apply it to find new exact solutions to nonlinear partial differential equations under proper initial conditions. Our solutions to nonlinear differential equations are given in series format, with convergence analysis for the proposed scheme. The new approach in this work can be considered as an alternative technique for the ones in the literature, to handle KdV-type equations such as the fifth-order Korteweg–de Vries (efKdV) equation, which is a crucial equation in fluid dynamics for the description of nonlinear wave processes. The new technique is a novel one to treat applications in the fields of mathematics, biology, engineering, physics, and other areas of science.

An efficient computational approach for fractional-order model describing the water transport in unsaturated porous media

This paper focuses on the application of an efficient technique, namely, the fractional natural decomposition method (FNDM). The numerical solutions of the model containing the water transport in unsaturated porous media, called Richards equation, are extracted. This model is used to describe the non-locality behaviors which cannot be modeled under the framework of classical calculus. To demonstrate the effectiveness and efficiency of the scheme used, two cases with time-fractional problems are considered in detail. The numerical stimulation is presented with results accessible in the literature, and corresponding consequences are captured with different values of parameters of fractional order. The attained consequences confirm that the projected algorithm is easy to implement and very effective to examine the behavior of nonlinear models. The reliable algorithm applied in this paper can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series.

Using the decomposition method to solve the fractional order temperature distribution equation: A new approach

Due to its importance in science, finding both exact and approximate solutions to fractional partial differential equations with boundary conditions is important for the research community. The natural decomposition method (NDM), which is based on the natural transformation method (NTM) and the Adomian decomposition method, is modified in this study to produce exact and approximate solutions for boundary value problems (BVPs) of partial differential equations (PDEs) with fractional coefficients. In addition, we present an exact solution to the temperature distribution in a slab constructed of materials with variable thermal conductivity's combined convection–radiation lumped system. We present these findings as numerical tables and graphs that show the convergence and stability rates. The study demonstrates that this approach is effective since it is simple to apply and produces reliable findings. We are the first to use this approach for such applications, as far as we are aware. Additionally, this method is applicable to a sizable class of BVPs for ordinary differential equations (ODEs) and PDEs.

Approximate Solutions of Time-Fractional Swift–Hohenberg Equation via Natural Transform Decomposition Method

Approximate Solutions of Time-Fractional Swift–Hohenberg Equation via Natural Transform Decomposition Method