Daftardar-Jafari Method Research Articles

Analysis of Natural Daftardar–Jafari Method for Fractional Delay Differential Equations

Analysis of Natural Daftardar–Jafari Method for Fractional Delay Differential Equations

A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation(WSVIE), $u(x)=f(x)+\int_{0}^{x}\frac{t^{\mu-1}}{x^\mu}[u(t)]^\beta dt$. AL-Jawary and Shehan used Daftardar-Jafari Method(DJM) and solved the above integral equation for the investigation parameter $\mu>1$ using specific force functions with $\mu$ and $\beta$ values and obtained unique solutions. We have discovered a force function $f(x)=x^{k_1}-{\frac{x^{\gamma k_1}}{\gamma k_1+\mu}}$, that allows the introduction of noise terms phenomena discovered by Wazwaz; that cancel out the terms of the power series in the successive solution terms $u_m$, $m=0,1,2,...,n$: we thus obtain a maximum finite power series terms for each solution term called truncation point and denoted by $x^{g(n)}$. Such that the integral solution can be written as $u(x)=u_0+\sum_{m=1}^{n}u_m$, where $n$ is finite. Simplifying the solution terms we get the unique solution $u(x)=x^{k_1}$, irrespective of $n-$value in the truncation point. We discovered a formula relation between the last solution term $u_n$ and the truncation point as $u_n=a_nx^{g(n)}$. Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter $\mu>1$. We extend the parameter range to include $\mu>1$ and $0<\mu\leq 1$ for our solution. In addition, for any chosen rational parameter $k_1$, the solution $u(x)=x^{k_1}$ is extrapolated to be valid for all integer parameter values $\beta\geq 2$, and positive rational parameter value $\mu>0$ and for any finite value of $n\geq2$.

DJM solution of MHD flow of ternary hybrid nanofluid between nonparallel a porous media channels with velocity slip and radiation effects

This investigation on heat transfer and Jeffery Hamel ternary hybrid nanofluid flow across nonparallel a porous media channels while accounting for radiation, velocity slip, and mixture fluid ( H 2 O − C 2 H 6 O 2 ) effects is presented in this work. The nonlinear PDEs such as continuity, momentum and heat equations transformed to a system of nonlinear ODEs using similarity transformations then solved numerically and analytically, the analytical solution has been constructed using by Daftar Dar–Jafari method (DJM), the present results in particular cases are compared to results obtained by the HAM-based Mathematica package and by the Runge–Kutta Fehlberg fourth–fifth order (RKF-45) for validation. The effects of active parameters such as the nanofluid volume fraction, velocity slip parameter, Darcy number (Da), and thermal radiation parameter (Rd), are investigated on the velocity and temperature, skin friction, and Nusselt numbers. It is found that the velocity of ternary hybrid nanofluid upsurges for both convergent and divergent channel with the increment in the Hartman number (Ha). Also, results obtained reveal that the temperature profile experiences an upward shift with an increase in the Ha, and a downward shift with an increment of the Da. Furthermore, new modeling and the reliable analytic treatment via DJM approach for the ternary hybrid nanofluid is a sizable accomplishment of the current analysis.

On the Semi-Analytical Solutions for the Kudryashov-Sinelshchikov Dynamical Equation Arising in Mixtures of Liquid and Gas Bubbles Without Neglecting of Heat Transfer and Viscosity

In this study, Banach contraction method (BCM), Daftardar-Jafari method (DJM) and modified variational iteration method (MVIM) are proposed for the semi-analytical solutions of the Kudryashov-Sinelshchikov (K-S) dynamical equation. It has been shown that the analytical and semi-analytical solutions for the K-S dynamical equation with initial value problems by using semi-analytical methods can be obtained. In addition, the effectiveness and usefulness of the semi-analytical methods used are supported by tables and 3D figures. As the number of iteration or terms increases, how the semi-analytical solutions behave over time and converge to the exact solution is shown in detail with 2D figures. Also, it is shown comparison of semi-analytical solutions with exact solutions and error analysis with the help of tables. It has been discussed the methods are compared with each other and whether they are suitable for the K-S dynamical equation.

A Review of Analytical Algorithms for Approximate Solutions of Ordinary Differential Equation

This review article aims to provide an in-depth analysis of a wide range of analytical methods used for solving ordinary differential equations (ODEs). ODEs are fundamental in numerous scientific and engineering disciplines, making the development, and understanding of effective solution techniques crucial. We explore various approaches, including the Adomian decomposition method, homotopy perturbation method, homotopy analysis method, variational iteration method, Daftardar-Jafari method, successive approximation method, power series method, and modified Adomian decomposition method. Each method is discussed in terms of its principles, applications, advantages, limitations, and computational considerations. This comprehensive overview will serve as a valuable resource for researchers, practitioners, and students interested in solving ODEs.

ANALYSIS OF THE CONFORMABLE TEMPORAL-FRACTIONAL SWIFT–HOHENBERG EQUATION USING A NOVEL COMPUTATIONAL TECHNIQUE

The main objective of this study is to provide a new computational procedure for extracting approximate and exact solutions of the temporal-fractional Swift–Hohenberg (S–H) equations in the context of conformable derivatives using the conformable natural transform (CNT) and Daftardar–Jafari method (DJM). We refer to it as the “natural conformable Daftardar–Jafari method” (CNDJM). The three types of errors are assessed in order to gauge the efficiency and consistency of the proposed method. Furthermore, 2D and 3D graphics are used to compare the exact and approximate solutions. This method offers a considerable benefit over homotopy analysis and Adomian decomposition methods in terms of computational work because it does not require Adomian and He’s polynomials. The procedure is quick and easy to use.

Light deflection angle in General Relativity via Daftardar-Jafari method

In this paper, the iterative method suggested by Daftardar and Jafari hereafter called Daftardar-Jafari method (DJM) is applied for studying the deflection of light in General Relativity. For this purpose, a brief review of the nonlinear geodesic equations in the spherical symmetry spacetime and the main ideas of DJM are given. As an illustrative example, the simple case of the Schwarzschild metric is considered for which the approximate solution to the null-geodesic equation and the deflection angle of light are obtained. We also compare the obtained result with some similar results presented earlier in the literature. Â

A novel computational approach for a nonlinear fractional model of plasma physics

In the current study, a well-known fractional nonlinear differential equation called Zakharov–Kuznetsov equation which has many applications in engineering and physics is investigated. An effective computational techniques namely Daftardar–Jafari Method (DJM) and He’s fractional derivative are employed to model the problem for obtaining a high accuracy of the analytical solution. The governing equations are solved and comparison of the results with exact solutions is presented to evaluate the precision of the presented method. The outcomes reveal that the DJM is efficient and easy to utilize. Further, three-dimensional contour plots are depicted according to the suitable parameters values.

Efficient Iterative Methods for Solving the SIR Epidemic Model

In this article, the numerical and approximate solutions for the nonlinear differential equation systems, represented by the epidemic SIR model, are determined. The effective iterative methods, namely the Daftardar-Jafari method (DJM), Temimi-Ansari method (TAM), and the Banach contraction method (BCM), are used to obtain the approximate solutions. The results showed many advantages over other iterative methods, such as Adomian decomposition method (ADM) and the variation iteration method (VIM) which were applied to the non-linear terms of the Adomian polynomial and the Lagrange multiplier, respectively. Furthermore, numerical solutions were obtained by using the fourth-orde Runge-Kutta (RK4), where the maximum remaining errors showed that the methods are reliable. In addition, the fixed point theorem was used to show the convergence of the proposed methods. Our calculation was carried out with MATHEMATICA®10 to evaluate the terms of the approximate solutions.

Iterative methods for approximate solution of the Ornstein-Uhlenbeck process with normalised brownian motion

This work considers the concept of the Normalised Brownian motion for the solutions of the Ornstein-Uhlenbeck process using the Daftardar-Jafari Method (DJM) and Picard Iterative Method (PIM) as the approximate-analytical methods of solutions. The results obtained from DJM are compared with those of the PIM. The obtained results, therefore, show the effectiveness of the proposed methods.

Doğrusal Olmayan Sınır Değerli Pantograf Tip Gecikmeli Diferansiyel Denklemlerin Nümerik Çözümleri

In this paper we compared the Daftardar-Jafari Method (DJM) with Adomian Decomposition Method (ADM) and Differential Transformation Method (DTM) in solving nonlinear boundary value delay differential equations of pantograph type. All these 3 methods provide series solutions to the problems. We analysed the first n-term approximate solutions of these 3 methods with 2 numerical examples to see if DJM is as good as ADM and DTM in solving nonlinear boundary value delay differential equations and we found DJM a reliable method in solving this kind of problems.

A Simple Analytic Approximation of Luminosity Distance in FLRW Cosmology using Daftardar-Jafari Method

In this paper, the iterative method suggested by Daftardar-Gejji and Jafari hereafter called Daftardar-Jafari method (DJM) is applied for the approximate analytical representation of the luminosity distance in a homogenous Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology. We obtain the analytical expressions of the luminosity distance using the approximate solutions of the differential equation to which the luminosity distance satisfies, subject to the corresponding initial conditions. With the help of this approximate solution, a simple analytic formula for the luminosity distance as a function of redshift is obtained and compared with a numerical solution for the general integral formula by the Maple software. Subsequent comparison of the obtained approximate analytical formula with the corresponding numerical solution for the and quintessential models is provided and showed a high accuracy of the DJM approximations, at least for the certain values of parameters of the models. This comparison demonstrates the efficiency and simplicity of this approach to the problem of calculating the luminosity distance in theoretical cosmology.

Analytic and numerical solutions for linear and nonlinear multidimensional wave equations

We develop three reliable iterative methods for solving the nonlinear 1D, 2D and 3D second-order wave equation and compare the results with a discretization-based solver. The iterative Tamimi–Ansari method (TAM), Daftardar–Jafari method (DJM) and the Banach contraction method (BCM) are used to obtain the exact solution for linear equations. For nonlinear equations and practical problems, however, one obtains the approximate solutions that converge to the exact solution, if one exists. The convergence analysis of the three methods is shown using the fixed-point theorem. The methods prove to be quite efficient and well suited to solve this kind of problems. We present several examples that demonstrate the accuracy and efficiency of the methods. We also compare the methods with a method based on discretization (Boundary Domain Integral Method (BDIM)). The BDIM uses a standard domain grid and discretizes the integral form of the governing equations. The iterative methods were developed with Mathematica® 10, while BDIM is a proprietary development.

Reliable iterative methods for mathematical model of COVID-19 based on data in Anhui, China

In this paper, five reliable iterative methods: Daftardar-Jafari method (DJM), Tamimi-Ansari method (TAM), Banach contraction method (BCM), Adomian decomposition method (ADM) and Variational iteration method (VIM) to obtain approximate solutions for a mathematical model that represented the coronavirus pandemic (COVID -19 pandemic). The accuracy of the obtained results is numerically verified by evaluating the maximum error remainder. In addition, the approximate results are compared with the fourth-order Runge-Kutta method (RK4) and good agreement have achieved. The convergence of the proposed methods is successfully demonstrated and mathematically verified. All calculations were successfully performed with MATHEMATICA®10.

Reliable iterative methods for 1D Swift–Hohenberg equation

In this paper, the nonlinear problem of the 1 D Swift-Hohenberg equation (S-HE) has been solved by using five reliable iterative methods. The first one is the Daftardar-Jafari method namely (DJM), the second method is the Temimi-Ansari method namely (TAM), the third method is the Banach contraction method namely (BCM), the fourth method is the Adomian decomposition method namely (ADM) and finally the fifth method is the Variational iteration method (VIM) to obtain the approximate solutions. In this work, we discussed and applied these iterative methods to solve the S-HE and compared them. In addition, the fixed-point theorem was given to illustrate the convergence of the five methods. To illustrate the accuracy and efficiency of the five methods, the maximum error remainder was calculated since the exact solution is unknown. The results showed that the five iterative methods are accurate, reliable, time saver and effective. All the iterative processes in this paper implemented in MATHEMATICA®11.

Reliable iterative methods for mathematical biology model

In this paper, three iterative methods will be used to solve the nonlinear differential equation systems that represents a model for HIV infection of cells. The iterative methods are the Daftardar-Jafari method namely (DJM), Temimi-Ansari method namely (TAM) and Banach contraction method namely (BCM) to get the approximate solutions. Numerical solutions were also obtained by using the Runge-Kutta (RK4) method and good agreement have achieved. In addition, the convergence of proposed methods is present based on Banach's fixed point theorem. Also, comparisons are made between the proposed methods and the Adomian decomposition method (ADM), and the Variational iteration method (VIM), a good agreement have achieved without required to evaluate the Adomian polynomials to handle the nonlinear terms as in the ADM, does not require to calculate Lagrange multiplier as in the VIM and time saver. Moreover, the results of the maximum error remainder values are obtained and show that the proposed methods are effective and reliable. Our calculations were performed using the MATHEMATICA®10.

Reliable Iterative Methods for Solving Convective Straight and Radial Fins with Temperature-Dependent Thermal Conductivity Problems

In our article, three iterative methods are performed to solve the nonlinear differential equations that represent the straight and radial fins affected by thermal conductivity. The iterative methods are the Daftardar-Jafari method namely (DJM), Temimi-Ansari method namely (TAM) and Banach contraction method namely (BCM) to get the approximate solutions. For comparison purposes, the numerical solutions were further achieved by using the fourth Runge-Kutta (RK4) method, Euler method and previous analytical methods that available in the literature. Moreover, the convergence of the proposed methods was discussed and proved. In addition, the maximum error remainder values are also evaluated which indicates that the proposed methods are efficient and reliable. Our computational works have been done by using the computer algebra system MATHEMATICA®10 to evaluate the terms in the iterative processes.

Analysis of various semi-numerical schemes for magnetohydrodynamic (MHD) squeezing fluid flow in porous medium

In this article comparative analysis of various semi-numerical schemes has been made for the case of squeezing flow of an incompressible viscous fluid between two large parallel plates having no-slip at the boundaries. The medium of flow contains magnetohydrodynamic (MHD) effect and having small pores. Modeled boundary value problem is solved analytically using Optimal homotopy asymptotic method (OHAM), homotopy perturbation method (HPM), differential transform method (DTM), Daftardar Jafari method (DJM) and Adomian decomposition method (ADM). For comparison purpose, residuals of these schemes have been found and analyzed for accuracy. Analytical study indicates that DTM and DJM are quite good in term of accuracy near the center of domain [−1, 1] but the accuracy reduces considerably near the start and end of the given interval. HPM and OHAM residuals indicate that OHAM surpasses HPM in terms of accuracy in the present case.

Iterative Methods For Solving Nonlinear Lane-Emden Equations]{Iterative Methods For Solving Nonlinear Lane-Emden Equations

In this paper we derive solutions for the nonlinear Lane-Emden type of equations with iterative methods, Daftardar-Jafari Method (DJM), Adomian Decomposition Method (ADM) and Differential Transform Method (DTM). The difficulty of Lane-Emden type of equations for implementing these iterative methods is due to the singularity at $x = 0$. We compare the efficiency of DJM - which is the authentic part of the paper - with ADM and DTM in solving the nonlinear, singular, initial value Lane-Emden type of equations.

Three iterative methods for solving second order nonlinear ODEs arising in physics

In this work, three iterative methods have been implemented to solve several second order nonlinear ODEs that arising in physics. The proposed iterative methods are Tamimi-Ansari method (TAM), Daftardar-Jafari method (DJM) and Banach contraction method (BCM). Each method does not require any assumption to deal with nonlinear term. The obtained results are compared numerically with other numerical methods such as the Runge-Kutta 4 (RK4) and Euler methods. In addition, the convergence of the proposed methods is given based on the Banach fixed point theorem. The results of the maximal error remainder values show that the present methods are effective and reliable. The software used for the calculations in this study was Mathematica®10.