A Residual Power Series Technique for Solving Systems of Initial Value Problems

Momani, Shaher; Arqub, Omar Abu; Hammad, Ma’mon Abu; Abo-Hammour, Zaer S.

doi:10.18576/amis/100237

Cited by 15 publications

(18 citation statements)

References 39 publications

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“…The modified homotopy analysis transform method (MHATM) method is a combination of the Laplace transform and homotopy analysis methods with HP [15][16][17][18]. The RPSM is constructed from the generalized Taylor series, which is a prevailing technique for solving nonlinear FDEs [19][20][21][22][23][24]. The advantage of the RPSM method is that it is not affected by computational round-off errors and also does not require large computer memory and extensive time.…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

Kumar

Kumar²,

Momani

et al. 2019

Adv Differ Equ

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The main aim of this paper is to present a comparative study of modified analytical technique based on auxiliary parameters and residual power series method (RPSM) for Newell–Whitehead–Segel (NWS) equations of arbitrary order. The NWS equation is well defined and a famous nonlinear physical model, which is characterized by the presence of the strip patterns in two-dimensional systems and application in many areas such as mechanics, chemistry, and bioengineering. In this paper, we implement a modified analytical method based on auxiliary parameters and residual power series techniques to obtain quick and accurate solutions of the time-fractional NWS equations. Comparison of the obtained solutions with the present solutions reveal that both powerful analytical techniques are productive, fruitful, and adequate in solving any kind of nonlinear partial differential equations arising in several physical phenomena. We addressed $L_{2}$ L 2 and $L_{\infty }$ L ∞ norms in both cases. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present aforesaid methods and noted excellent agreement. In this study, we use the fractional operators in Caputo sense.

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Section: Introductionmentioning

confidence: 99%

Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

Kumar

Kumar²,

Momani

et al. 2019

Adv Differ Equ

Self Cite

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“…where For numerical results and comparisons, the following values, for parameters, are considered: = 0.05, 0 = 1200, 0 = 25, and 1 = 2 = 0.5. The exact fuzzy solutions of system of FIVP (13) and (14)…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Solutions of the FDEs can often be expressed in terms of series expansions. However, the RPS technique is an analytical as well as numerical method for solving different types of ordinary and partial differential equations, integral equation and integro-differential equation [7][8][9][10][11][12][13][14][15][16]. The methodology is effective and easy to construct power series solution for strongly linear and nonlinear systems of FIVPs without linearization, perturbation, or discretization [17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

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Analytical treatment for solving system of fuzzy IVPs using residual power series approach

Aloroud¹

2017

imf

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The aim of the present analysis is present a relatively new analytical treatment, called residual power series (RPS) method, for solving system of fuzzy initial value problems under strongly generalized differentiability. The technique methodology provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. Several computational experiments are given to show the good performance and potentiality of the proposed procedure. The results reveal that the present simulated method is very effective, straightforward and powerful methodology to solve such fuzzy equations.

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“…Generally, partial differential equations (PDEs) are hard to tackle, and their fractional-order types are more complicated [7,8]. Therefore, several analytical and approximate methods can be used for finding their approximate solutions such as Adomian decomposition [9], homotopy analysis [10], tau method [11], residual power series method [12], and optimal homotopy asymptotic method [13]. Though the study of FPDEs has been obstructed due to the absence of proficient and accurate techniques, the derivation of approximate solution of FPDEs remains a hotspot and demands to attempt some dexterous and solid plans which are of interest.…”

Section: Introductionmentioning

confidence: 99%

New Iterative Method for the Solution of Fractional Damped Burger and Fractional Sharma‐Tasso‐Olver Equations

et al. 2018

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The new iterative method has been used to obtain the approximate solutions of time fractional damped Burger and time fractional Sharma-Tasso-Olver equations. Results obtained by the proposed method for different fractional-order derivatives are compared with those obtained by the fractional reduced differential transform method (FRDTM). The 2nd-order approximate solutions by the new iterative method are in good agreement with the exact solution as compared to the 5th-order solution by the FRDTM.

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A Residual Power Series Technique for Solving Systems of Initial Value Problems

Cited by 15 publications

References 39 publications

Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

Analytical treatment for solving system of fuzzy IVPs using residual power series approach

New Iterative Method for the Solution of Fractional Damped Burger and Fractional Sharma‐Tasso‐Olver Equations

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