Adomian Decomposition Method Research Articles

Solving Fuzzy Delay Differential Equations using Fuzzy Elzaki Transform and Fuzzy Elzaki Decomposition Method

Objective: In this work, the fuzzy Elzaki transform and fuzzy Elzaki decomposition method are used to solve fuzzy delay differential equations. The fuzzy Elzaki decomposition method is a combination of the fuzzy Elzaki transform and Adomian decomposition method in a fuzzy environment. The use of transforms in solving differential equations makes the solution process simpler. Methodology: The fuzzy Elzaki transform is applied in the fuzzy delay differential equation. The nonlinear terms are decomposed using Adomian polynomials in a nonlinear fuzzy delay differential equation. In the transformed domain, the resulting algebraic equation is solved. The solution in the original time domain is obtained by applying the inverse Elzaki transform. Findings: The fuzzy Elzaki transform and fuzzy Elzaki decomposition method are useful for handling the complexities associated with fuzzy functions and delays in differential equations. The technique used to solve the fuzzy delay differential equation gives results with better accuracy compared to the exact solution. The applicability of the method is illustrated with numerical examples. Novelty: The Elzaki transform is a helpful tool for solving delay differential equations with discontinuities. For the resilient solution of complicated differential equations, especially those including nonlinearity and fuzzy logic, the fuzzy Elzaki decomposition approach is useful. It gives an organized way to find the solution to fuzzy delay differential equation by utilizing the advantages of both the Elzaki transform and the Adomian decomposition method in a fuzzy situation. Keywords: Triangular Fuzzy Number, Fuzzy Elzaki Transform, Adomian decomposition method, Fuzzy Elzaki decomposition method, Fuzzy Delay Differential Equations (FDDE)

A Novel Approach to Nonlinear Volterra-Fredholm Integral Equations Using Abaoub Shkheam Decomposition Method

In this study, we introduce a novel approach to the solution of a nonlinear Volterra -Fredholm integral equations by applying the Adomian decomposition method under the effect of the Abaoub- Shkheam transform. We demonstrate the existence and uniqueness of the solution in Banach space and illustrate this idea with an example.

Advancing Non-Linear PDE’s Solutions: Modified Yang Transform Method with Caputo-Fabrizio Fractional Operator

Abstract In many physical processes, nonlinear PDEs have a crucial role. This paper presents a novel approach named as Modified Yang Transform (MYT) that combines an integral transform with the decomposition technique. This approach uses Yang transform and Adomian decomposition method to solve nonlinear fractional order PDEs. Nonlinear PDEs of fractional order are considered in the Caputo-Fabrizio sense. This work presents Convergence analysis of the modified Yang transform to nonlinear fractional order PDEs. Additionally, a solution framework to solve the nonlinear PDEs is carried out and some examples are provided to highlight the application of the current method.
To illustrate how the solution operates for various fractional orders, 2D and 3D graphs are plotted. After each example, the results and discussion section is included to explain the graphs and their results.

Reliable Computational Method for Systems of Fractional Differential Equations Endowed with ψ-Caputo Fractional Derivative

This study develops a highly convergent computational method, the ψ-Laplace Adomian Decomposition Method (ψ-LADM), for solving coupled systems ofψ-Caputo Fractional Differential Equations (FDEs). The effectiveness of the proposed method has been assessed using various numerical test examples, including a real-world application for atmospheric convection models utilizing Lorenz chaotic dynamical systems. Notably, the method consistently produced solutions that matched the true solutions of the governing models. In the case of the Lorenz chaotic system, the obtained solutions accurately portrayed the characteristic phase portraits of a true chaotic system.

Simulation of blood flow in a tapered artery with ternary hybrid nanofluid and Jeffrey model

This research simulates flow of blood in a tapered peristaltic artery integrating ternary hybrid nanofluid using the Jeffrey fluid model. The simulation examines the effects of Au, Fe3O4, and Ag nanoparticles on the flow. The artery, modeled as a peristaltic channel, is subjected to nonlinear thermal radiation, magnetic forces, and heat source. The problem is formulated using non-linear Cartesian partial differential equations, which are then transformed into dimensionless form without approximations. Adomian Decomposition Method (ADM) is employed to solve these equations, revealing how normal and shear stress, normal and axial velocities, induced magnetic fields, and temperature profiles vary with changes in relevant parameters. Additionally, biological factors are considered to graph streamlines, providing a comprehensive analysis of the flow dynamics. Some notable results include that adding ternary nanoparticles increases the Nusselt number by 26.03% in Newtonian fluids and 158.69% in Jeffrey fluids, and increasing tapering parameters and wavenumber improves axial and normal velocities, heat transfer, and flow complexity.

Generalized Laplace Transform with Adomian Decomposition Method for Solving Fractional Differential Equations Involving ψ-Caputo Derivative

In this study, we introduced the ψ-Laplace transform Adomian decomposition method, which is a combination of the efficient Adomian decomposition method with the generalization of the classical Laplace transform to treat fractional differential equations with respect to another function, ψ, in the Caputo sense. To validate the effectiveness of this method, we applied the derived recurrent scheme of the ψ-Laplace Adomian decomposition on several test numerical problems, including a real-life scenario in pharmacokinetics that models the movement of drug concentration in human blood. The solutions obtained closely matched the known solutions for the test problems. Additionally, in the pharmacokinetics case, the results were consistent with the available physical data. Consequently, this method simplifies the verification of numerous related aspects and proves advantageous in solving various ψ-fractional differential equations.

Enhancement efficiency of flow and irreversibility system for MHD Buongiorno's nanofluid in complex peristaltic tapered channel with electroosmosis forces

Abstract Magnetohydrodynamic flow efficiency and irreversibility improvement research are multiple problems that arise when electroosmosis forces affect Buongiorno's nanofluid in a complicated peristaltic tapered channel. Thermal energy and temperature gradients cause nanoparticles to migrate randomly, affecting flow efficiency and irreversibility. Sometimes the Infected veins generate complex peristaltic waves on its walls. The mathematical model that characterizes the motion of Jeffrey magnetohydrodynamic Buongiorno's nanofluid inside a complex tapered peristaltic channel, considering the effects of electroosmotic forces is discussed. The long wavelength and low Reynolds numbers approximation is considered. The approximate solution of the nonlinear system of partial differential formulas is obtained using the Adomian decomposition method. Also, the irreversibility of the system and entropy generation are being studied. Flow characteristics with biophysical and thermal parameters are plotted and discussed. The improvement in the interstitial distances between the molecules that make up the nanofluid, which in turn works to enhance the Bejan numbers. So, one important result is by growing in both Brownian motion and thermophoresis, the Bejan numbers rise clearly. Both the Jeffrey parameter and Debye–Huckel parameter work to upsurge the loss of kinetic energy within the molecules, which reduces the temperatures inside the nanofluid and thus reduces the entropy rate, in contrast to the rest of the parameters that raise the kinetic energy inside the molecules that make up the nanofluid.

Enhanced Adomian Decomposition Method for Accurate Numerical Solutions of PDE Systems

This research addresses critical challenges in numerical solutions, which are vital for various engineering and physical fields. The Modified Adomian Decomposition Method (MADM) is proposed as a novel approach for solving linear and nonlinear partial differential equations (PDEs). MADM builds upon the Adomian Decomposition Method (ADM) by incorporating a new integral operator that significantly improves convergence rates and accuracy. Numerical examples demonstrate the effectiveness of MADM in handling complex nonlinear PDEs. Compared to traditional ADM, MADM consistently achieves more accurate and rapidly converging solutions. This enhancement is attributed to the novel integral operator, which addresses the limitations of ADM for intricate nonlinear problems. The paper outlines the application of MADM, its solution procedure, and its effectiveness through numerical examples. Comparisons with standard ADM solutions and exact solutions validate MADM's accuracy and superiority. The results suggest that MADM is a promising tool for expanding the applicability of Adomian methods in the field of solving PDEs.

Theoretical analysis of facilitated diffusion process in a liquid membrane: Adomian decomposition method

The mathematical model of facilitated transport systems in steady-state conditions is discussed. The nonlinear component associated with chemical reactions and diffusion in nonlinear equations forms the basis of this model. By solving the nonlinear equation using the Adomian decomposition method, we can obtain the analytical expression of the concentration of species. The simple expression of the facilitation factor ( ratio between the total flux of solute and that of the absence of carrier) is also reported. The parameters' influence on species concentration and facilitation factors are discussed. A good agreement is obtained when we compare the numerical results for all parameter values with new analytical results.

Multi-step Adomian decomposition method for solving a delayed Chikungunya virus system

Multi-step Adomian decomposition method for solving a delayed Chikungunya virus system

Shape factor and thermal radiation effects on convective MHD flow of Casson blood ternary hybrid nano‐fluid through a stretched rotating rigid disk

AbstractThe mathematical modeling of blood flow with the incorporation of ternary hybrid nanoparticles (i.e., titania, silica, and alumina nanoparticles) is the focus of this work. The flow of blood is simulated using the Casson fluid model. The impacts of ternary hybrid nanoparticles, shape factor, and Geometry of solid nanoparticles are visualized and investigated. The momentum and thermal characteristics of a flowing liquid are determined by considering magnetization, porosity, and nonlinearized radiating thermal flux. The basic partial differential equations resulting from mathematical modeling are turned into non‐linear ordinary differential equations using suitable velocity transforms. The derived nonlinear equations are then numerically calculated using the 4th‐5th order Runge‐Kutta‐Fehlberg method with the shooting technique and analytically solved by the Adomian decomposition method (ADM). The effects of the factors involved on the dimensionless profiles produced are fully addressed. It is found that the Skin friction factor and heat transfer rate in the radial direction upsurge with the augment of both rotation parameter and nanoparticles volume fraction; however, the Skin friction factor drops in the azimuthal direction. Also, results obtained reveal an enhancement in the local Nusselt number with the upsurge in the magnitude of radiation parameter, Rd, solid nanoparticles concentration, , shape factor value, s, and disk temperature, . For validation, the outcomes of this inquiry were compared to the outcomes of the HAM‐based Mathematica software. In addition, the acquired analytical DRA data are compared to numerical RKF45 values and those given in the literature.

Analytical solutions for the Noyes Field model of the time fractional Belousov Zhabotinsky reaction using a hybrid integral transform technique

In this work, we employed an attractive hybrid integral transform technique known as the natural transform decomposition method (NTDM) to investigate analytical solutions for the Noyes-Field (NF) model of the time-fractional Belousov–Zhabotinsky (TF-BZ) reaction system. The aforementioned time-fractional model is considered within the framework of the Caputo, Caputo–Fabrizio, and Atangana–Baleanu fractional derivatives. The NTDM couples the Adomian decomposition method and the natural transform method to generate rapidly convergent series-type solutions via an elegant iterative approach. The existence and uniqueness of solutions for the considered time-fractional model are first investigated via a fixed-point approach. The reliability and efficiency of the considered solution method is then demonstrated for two test cases of the TF-BZ reaction system. To demonstrate the validity and accuracy of the considered technique, numerical results with respect to each of the mentioned fractional derivatives are presented and compared with the exact solutions as well as with those from existing related literature. Graphical representations depicting the dynamic behaviors of the chemical wave profiles of the concentrations of the intermediates are presented with respect to varying fractional parameter values as well as temporal and spatial variables. The obtained results indicate that the execution of the method is straightforward and can be employed to explore nonlinear time-fractional systems modeling complex chemical reactions.

The extended Adomian decomposition method and its application to the rotating shallow water system for the numerical pulsrodon solutions

The rotating shallow water system is an important physical model, which has been widely used in many scientific areas, such as fluids, hydrodynamics, geophysics, oceanic and atmospheric dynamics. In this paper, we extend the application of the Adomian decomposition method from the single equation to the coupled system to investigate the numerical solutions of the rotating shallow water system with an underlying circular paraboloidal basin. By introducing some special initial values, we obtain interesting approximate pulsrodon solutions corresponding to pulsating elliptic warm-core rings, which take the form of realistic series solutions. Numerical results reveal that the numerical pulsrodon solutions can quickly converge to the exact solutions derived by Rogers and An, which fully shows the efficiency and accuracy of the proposed method. Note that the method proposed can be effectively used to construct numerical solutions of many nonlinear mathematical physics equations. The results obtained provide some potential theoretical guidance for experts to study the related phenomena in geography, oceanic and atmospheric science.

Adomian decomposition method for modelling the growthof FeB/Fe2B layer in boronizing process

The main objective of this paper is to explore the practical implementation of the Adomian decomposition method (ADM) in effectively solving the system of equations governing boron diffusion during the boronizing process. This study uses ADM to investigate the kinetics of the boronizing process, assess the influence of various parameters on the growth of the layer thickness, and determine the boron concentration in FeB and Fe2B phases. To validate the simulation results, data obtained from the literature were utilized. Overall, this research contributes to understanding the boronizing process and demonstrates the effectiveness of ADM as a mathematical tool for solving complex diffusion equations.

Revealing the dynamics of stagnant rings of third-grade fluid film with heat transfer in the presence of surface tension

In many engineering applications, including coating and lubrication operations, analyzing the temperature behavior of thin film flows on a vertically upward-moving tube is crucial to improving predictive models. This paper examines a steady third-grade fluid film flow with a surface tension gradient on a vertical tube. The mechanisms responsible for the fluid motion are upward tube motion, gravity, and surface tension gradient. This analysis focuses on heat transfer and stagnant ring dynamics. The formulated highly nonlinear ordinary differential equations are solved using the Adomian decomposition method. The conditions for stagnant rings and uniform film thickness are attained and discussed. The inverse capillary number C, Stokes number St, Deborah number De, and Brinkman number Br emerged as flow control parameters. The temperature of the fluid film rises with an increase in the C, St, De, and Br, whereas it decreases with an increase in thermal diffusion rate. The radius of stagnant rings tends to shrink by the increase in C, St, and De. When the value of De is high, third-grade fluid behaves like solids; only free drainage happens with smaller radius stagnant rings and high temperatures. A comparison between Newtonian and third-grade fluids regarding surface tension, velocity, temperature, stationary rings, and fluid film thickness is also provided.

Mass superinflation in the Reissner-Nordström black hole

Ever since Penrose and Simpson's prediction contradicted Novikov's idea that an infalling person would emerge into an asymptotically flat universe, there have been a continued interest in predicting the nature of singularity at the inner horizon of a Reissner-Nordström black hole. This prediction was first confirmed by Poisson and Israel using cross-stream of massless particles, leading to the phenomenon coined as mass inflation. On the other hand, Ori obtained a weaker singularity using a null shell of radiation. Thus it is important to capture the nature of singularity at the inner horizon. We therefore consider in this work a massive scalar field coupled to the Reissner-Nordström spacetime. The ensuing field equations lead to a coupled set of nonlinear dynamical equations. For definiteness, we analytically solve these equations employing the Adomian decomposition method. This facilitates in obtaining a closed form solution that exhibits an unbounded double-exponential growth in the mass function, giving rise to a novel phenomenon coined herein as mass superinflation. The scalar field is also found to undergo a very strong blueshift at the inner horizon.

A Variant of Accelerated Ramadan Group Adomian Decomposition Method for Numerical Solution of Fractional Riccati Differential Equations

A Variant of Accelerated Ramadan Group Adomian Decomposition Method for Numerical Solution of Fractional Riccati Differential Equations

Solving Undamped and Damped Fractional Oscillators via Integral Rohit Transform

Background: The dynamics of fractional oscillators are generally described by fractional differential equations, which include the fractional derivative of the Caputo or Riemann-Liouville type. These equations induce classical oscillator equations like the harmonic oscillator equation, to include fractional order derivatives. Solving fractional differential equations numerically can be challenging due to the non-local nature of fractional derivatives. Objective: In this paper, a recently developed integral Rohit transform is utilized for solving systems of undamped and damped fractional oscillators characterized by differential equations of fractional or non-integral order involving the Caputo-fractional derivative operator. The solutions of fractional systems which include undamped-simple fractional oscillators, undamped-driven fractional oscillators, damped-driven fractional oscillators, and damped-fractional oscillators are obtained. Methods: by applying the integral Rohit transform, also written as RT. Differential equations of fractional or non-integral order are generally solved by utilizing methods which include the fractional variational iteration approach, the homotopy-perturbation method, the equivalent linearized method, the Adomian decomposition method, etc. Results: This paper demonstrates the effectiveness, reliability, and efficiency of the integral Rohit transform in solving fractional systems, which include undamped-simple fractional oscillators, undamped-driven fractional oscillators, damped-driven fractional oscillators, and damped-fractional oscillators and are characterized by differential equations of fractional or non-integral order involving the Caputo-fractional derivative operator. Conclusions: The Rohit transform brought the progressive principles or methodologies that offer new insights or views on the problems examined in the paper, distinguishing itself from existing methods and doubtlessly beginning up new research instructions. It provided precise results for the specific problems discussed in the paper, surpassing the capabilities of other methods in terms of decision, constancy, or robustness to noise and disturbances.

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

Entropy analysis of Williamson fluid flow in a vertical porous channel with asymmetric cooling: Adomian decomposition method

AbstractThe paper presents the execution of the Adomian Decomposition Method to treat thermo‐fluidics of Williamson fluid flow in a vertical parallel plates channel filled with Williamson fluid–saturated porous medium. The channel walls experience asymmetric cooling. The fully developed flow setup constitutes a non‐linear boundary value problem tackled by the Adomain Decomposition Method (ADM) with the help of MAPLE. The computational procedure yields velocity, temperature distributions, skin friction, and Nusselt number. The results thus obtained are compared with those obtained by the RK‐Fehlberg numerical method to ascertain the accuracy of ADM. The results for quantities of interest viz. Skin friction. Nusselt number, entropy, and global entropy are shown in tabular/pictorial forms. These quantities exhibit qualitative and quantitative responses to embedded parameters discussed in detail in results and discussions section.