Exact solutions of Laplace equation by DJ method

Yaseen, Muhammad; Samraiz, Muhammad; Naheed, Saima

doi:10.1016/j.rinp.2013.01.001

Cited by 15 publications

(4 citation statements)

References 12 publications

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“…This iterative method has been successfully used to solve many kinds of problems. For instance; the application of DJM for solving different kinds of partial differential equations (Bhalekar & Daftardar-Gejji, 2008, 2012Daftardar-Gejji & Bhalekar, 2010), solving the Laplace equation (Yaseen et al, 2013), solving the Volterra integro-differential equations with some applications for the Lane-Emden equations of the first kind (AL-Jawary & AL-Qaissy, 2015), solving the Fokker-Planck equation (AL-Jawary, 2016), Duffing equations (Al-Jawary & Al-Razaq, 2016) and calculating the steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions (Al-Jawary & Raham, 2016), and others. The thin film flow problem has been solved previously by the ADM and VIM (Siddiqui, Farooq, Haroon, & Babcock, 2012a), the semi-analytical iterative method by Temimi and Ansari (TAM) (AL-Jawary, 2017) and other known iterative methods (Gul, Islam, Shah, Khan, & Shafie, 2014;Mabood, 2014;Mabood & Pochai, 2015;Moosavi, Momeni, Tavangar, Mohammadyari, & Rahimi-Esbo, 2016;Nemati, Ghanbarpour, Hajibabayi, & Hemmatnezhad, 2009;Sajid & Hayat, 2008;Shah, Pandya, & Shah, 2016;Siddiqui, Farooq, Haroon, Rana, & Babcock, 2012b).…”

Section: Introductionmentioning

confidence: 99%

Daftardar-Jafari method for solving nonlinear thin film flow problem

AL-Jawary

Radhi

Ravnik

2018

Arab Journal of Basic and Applied Sciences

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The aim of this paper is to develop the Daftardar-Jafari iterative method (DJM) for a mathematical model that represents the nonlinear thin film flow of a non-Newtonian third-grade fluid on a moving belt with the aim to obtain an approximate solution of high accuracy. When applying the DJM there is no need to resort to any additional techniques such as evaluating Adomian's polynomials as in the Adomian decomposition method (ADM) or such as using Lagrange multipliers in the variational iteration method (VIM). The accuracy of our results is numerically verified by evaluating the functions of the error remainder and the maximal error remainders. In addition, these results are analyzed by comparing the accuracy of the DJM solutions with those of the fourth order Runge-Kutta method (RKM), ADM and VIM at the same parameter values. All the evaluations have been successfully performed in an iterative way by using the symbolic manipulator Mathematica V R .

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Section: Introductionmentioning

confidence: 99%

Daftardar-Jafari method for solving nonlinear thin film flow problem

AL-Jawary

Radhi

Ravnik

2018

Arab Journal of Basic and Applied Sciences

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“…Jamil et al [16] using differential transform could find a closed-form solution for Laplace equation with Dirichlet and Neumann boundary conditions which in comparison with iterative solutions is quicker. Yaseen et al [17] employed a modified DJ method, and presented an iterative method for solving Laplace problem in Dirichlet and Neumann boundary conditions and rectangular environment. Ji-Huan et al [18] modified an iterative vibrational method for approximate solution of nonlinear differential equations.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solution of Laplace and Poisson Equations Using Conformal Mapping and Kronecker Products

2016

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In this paper, using the eigenvalues and eigenvectors of symmetric block diagonal matrices with infinite dimension and numerical method of finite difference, a closed-form solution for exact solution of Laplace equation is presented. The method of this paper has applications in different states of boundary conditions like Neumann, Dirichlet, and other mixed boundary conditions. Using the method of this paper, a mathematical model for the exact solution of the Poisson equation is derived. Since these equations have many applications in engineering problems, in each part of this paper, examples, like water seepage problem through the soil and torsion of prismatic bars, are presented. Finally, a method is provided for torsion problem of prismatic bars with non-circular and non-rectangular cross-sections utilizing conformal mapping.

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“…The DJ method has been extensively used by many researchers for the treatment of linear and nonlinear ordinary and partial differential equations of integer and fractional order [9,10,11,12]. The method converges to the exact solution if it exists through successive approximations.…”

Section: Introductionmentioning

confidence: 99%

A reliable iterative method for solving the epidemic model and the prey and predator problems

AL-Jawary

2014

IJBAS

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In the present article, we implement the new iterative method proposed by Daftardar-Gejji and Jafari (NIM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] to solve two problems; the first one is the problem of spread of a non-fatal disease in a population which is assumed to have constant size over the period of the epidemic, and the other one is the problem of the prey and predator. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM), does not require to calculate Lagrange multiplier as in Variational Iteration Method (VIM) and no needs to construct a homotopy as in Homotopy Perturbation Method (HPM). The results obtained are compared with the results by existing methods and prove that the presented method is very effective, simple and does not require any restrictive assumptions for nonlinear terms. The software used for the numerical calculations in this study was MATHEMATICA 8.0.

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Exact solutions of Laplace equation by DJ method

Cited by 15 publications

References 12 publications

Daftardar-Jafari method for solving nonlinear thin film flow problem

Daftardar-Jafari method for solving nonlinear thin film flow problem

Analytical Solution of Laplace and Poisson Equations Using Conformal Mapping and Kronecker Products

A reliable iterative method for solving the epidemic model and the prey and predator problems

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