Convergence of Discrete Adomian Method for Solving a Class of Nonlinear Fredholm Integral Equations

Alkalla, Ibrahim Lotfy Hassan; Abd-Elmonem, Reda Abdo; Gomaa, Ayman Mohamed Ali Ahmed

doi:10.4236/am.2013.41a033

Cited by 2 publications

(5 citation statements)

References 22 publications

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“…It is easy to prove (Ι ) and (ΙΙ ) as the same manner in (Al-Saif, 2015;Alkalla et al, 2013;Inc, 2005) ). Hence the prove is complete.…”

Section: Proofmentioning

confidence: 97%

“…In this section, we study the analysis of convergence in the same manner as (Alkalla et al, 2013;Inc, 2005;Jang, 2007) for the decomposition method to the nonlinear 1D compressible NavierStokes Eqs. 18-19.…”

Section: Convergence Analysis Of Sdhpmmentioning

confidence: 99%

“…Now, we will use the following theorem to satisfy the above conditions as (Alkalla et al, 2013;Inc, 2005). Proof: It is easy to prove (Ι ) and(ΙΙ ) as the same manner in (Al-Saif, 2015;Alkalla et al, 2013;Inc, 2005) to obtain on the results: Then condition (Ι ) holds with 1 = − (1 − 3 ), where 3 are constant and the condition (ΙΙ ) is satisfied with ( ) = − (1 − 4 ), where 4 is constant.…”

Section: Test Problem (P2) 2dmentioning

confidence: 99%

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A new approach for solving compressible Navier-Stokes equations

Al-Saif¹,

Al‐Griffi

2017

Int. j. adv. appl. sci.

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In this paper, we introduce the analytical approximate solutions for one and two-dimension compressible Navier-Stokes equations by applying a relatively new method named splitting decomposition homotopy perturbation method. The new methodology depends on combining Adomian decomposition and Homotopy perturbation methods with the splitting time scheme for differential operators. The numerical results which we obtained from the solutions of the two problems, show that the new method is efficient with good converge and high accuracy compared with the two standard methods i.e. Adomian decomposition method and Homotopy perturbation method.

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“…It is easy to prove (Ι ) and (ΙΙ ) as the same manner in (Al-Saif, 2015;Alkalla et al, 2013;Inc, 2005) ). Hence the prove is complete.…”

Section: Proofmentioning

confidence: 97%

Section: Convergence Analysis Of Sdhpmmentioning

confidence: 99%

Section: Test Problem (P2) 2dmentioning

confidence: 99%

See 1 more Smart Citation

A new approach for solving compressible Navier-Stokes equations

Al-Saif¹,

Al‐Griffi

2017

Int. j. adv. appl. sci.

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show abstract

“…Now , we will use the following theorem to satisfy the above conditions as [35,36] Theorem 1: If    and ) ( are satisfied , then ADM of Equation (17) Proof : It is easy to prove (I) and (II) as the same manner in [6,35,36] …”

Section: -Analysis Of Convergence Of Sdhpmmentioning

confidence: 99%

“…In this section, we will study the analysis of convergence in the same manner as [35,36,37] for the decomposition method to the nonlinear Navier-Stokes Equation (6). Let as consider the Hilbert space H which may be defined as…”

Section: -Analysis Of Convergence Of Sdhpmmentioning

confidence: 99%

Splitting Decomposition Homotopy Perturbation Method To Solve One -Dimensional Navier -Stokes Equation

Al-Saif

Al‐Griffi

2017

JAM

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I S S N 2347-1921 V o l u m e 13 N u m b e r 2 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 7123 | P a g e A p r i l 2 0 1 7 w w w . c i r w o r l d . c o mSplitting decomposition homotopy perturbation method to solve onedimensional Navier-Stokes equation 1-IntroductionNavier-Stokes equations are non-linear partial differential equations that govern the incompressible fluid flow inside specific hollow as it is an equation of parabolic type and the non-linear degree is increased with the increase of Reynolds number . These equations names are taken from the two physicists Claude-Louis Navier and George Gabriel Stokes in the nineteenth century, and resulted from applying Newton second law on the movement of fluids supposing that stress of fluid is totaled of divergence of viscosity (proportional with velocity rate) and pressure. Also, these equations are considered as the most important physical equations which describe a large number of phenomena of different applications in many research fields that may be used in modeling weather, liquid flow in channels and pipes , gas flow round flying bodies, and movement of stars in the galaxy.Navier-Stokes equations are important mathematically, due to their wide applications, where to this day no one has succeeded in proving the existence of general solution or a permanent for Navier-Stokes equations( privately threedimensional). Therefore, this kind of problems is named as the existence and smooth flow of Navier-Stokes. During few last years, many researchers attempted to modify and extend these two methods; for example, Zhang [14 ] modified ADM to solve a class of non-linear boundary value problems. Jafari and Daftardar-Gejji [15] modified ADM to solve a set of non-linear equations led to find approximation solutions are better than those which were obtained by standrad method in measurements of convergent and accuracy. Luo [16] suggested active methods for ADM which is a two-steps Adomian decomposition method (TSADM) to reach the solution. TSADM reduces the repetitive of mathematical processes that are applied to find the solution and also he made comparison for the results. The results showed that TSADM is an active and efficient method which has high accuracy in finding solutions. Also, in many works [17,18,19,20], the authors used ADM to found analytic and approximate solutions for different problems. In the same direction of modification, the HPM is active to find solutions for non-linear equations [10,[21][22][23][24][25][26][27][28][29][30].Depending on the above literature review for attempt of researchers to expand and developed ADM and HPM to solve linear and nonlinear boundary value problems, and depending on our simple information about applications of these methods to solve the problems that under consideration study, we did not find, if any, that is for problems are different from our problem in formulations and conditions. These information encourage us to suggest a new methodology for solving nonlinear problems. The new methodology is con...

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Convergence of Discrete Adomian Method for Solving a Class of Nonlinear Fredholm Integral Equations

Cited by 2 publications

References 22 publications

A new approach for solving compressible Navier-Stokes equations

A new approach for solving compressible Navier-Stokes equations

Splitting Decomposition Homotopy Perturbation Method To Solve One -Dimensional Navier -Stokes Equation

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