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Ghawadri, Nizam; Senu, Norazak; Ismail, Fudziah; İbrahim, Zarina Bibi

doi:10.1155/2018/4029371

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…In this section, the APM performance will be tested on a well-known application in engineering and physics (Thin Film Flow). The thin film flow of a liquid on a certain solid surface problem has received a lot of attention [29,30].…”

Section: Application To Thin Film Flow Problem Using Apmmentioning

confidence: 99%

“…Considering the cases, where x1 = x2 = x3 = 1, k = 2, and k = 3, the proposed APM is also applied to analytically solve (37). The results are summarized in Table V and Table VI and compared with the results of all other numerical methods [29,30], for the case of k = 3 and the analytical method in [16] for the case of k = 2.…”

Section: Application To Thin Film Flow Problem Using Apmmentioning

confidence: 99%

“…The absolute values of the computed errors between the best-estimated numerical solutions [29,30] and those of the proposed APM, at the discretized points of interest, compared to the closed-form solution are shown in Fig. 12.…”

Section: Application To Thin Film Flow Problem Using Apmmentioning

confidence: 99%

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Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

2021

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Differential equations are commonly used to model several engineering, science, and biological applications. Unfortunately, finding analytical solutions for solving higher-order Ordinary Differential Equations (ODEs) is a challenge. Numerical methods represent a leading candidate for solving such ODEs. This work presents an innovated adaptive technique that uses polynomials to solve linear or nonlinear third-order ODEs. The proposed technique adapts the coefficients of the polynomial to obtain an explicit analytical solution. A signed least mean square algorithm is exploited to enhance the adaptation process and to decrease both computational requirements and time. The efficiency of the proposed Adaptive Polynomial Method (APM) is illustrated through six well-known examples. The proposed technique is compared with recent analytical and numerical methods to validate its effectiveness in terms of Mean Square Error (MSE) and computation time. An application in a thin film flow system is modeled to a third-order ODE. The proposed technique is compared with recent numerical and analytical methods in solving the thin film flow equation, and the proposed technique has achieved better results. Furthermore, the proposed technique provides an analytical solution with an increased dynamic range and much lower computational time than those of the conventional numerical methods.

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Section: Application To Thin Film Flow Problem Using Apmmentioning

confidence: 99%

Section: Application To Thin Film Flow Problem Using Apmmentioning

confidence: 99%

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Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

2021

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“…Numerous problems in mathematics can be formulated in the form of differential equations, an initial value problem (IVP) is an ordinary differential equation (ODE) whose boundary conditions are specified at a single point, which can be found in mathematical modeling of real-life problems [1]- [3]. There is also another class of the ODE which is the boundary value problem (BVP), a BVP differs from an IVP in that the boundary conditions are specified at more than one point and in that solutions of the differential equation over an interval, satisfying the boundary conditions at the endpoints, are required ( [4], p.1).…”

Section: Introductionmentioning

confidence: 99%

Direct One-Step Method for Solving Third-Order Boundary Value Problems

Abdulsalam¹,

Senu²,

Majid³

2019

Int. J. of Appl. Math.

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A direct explicit Runge-Kutta type (RKT) method via shooting technique to approximate analytical solutions to the third-order two-point boundary value problems (BVPs) with boundary condition type I and II are proposed. In this paper first, a three-stage fourth-order direct explicit Runge-Kutta type method denoted as RKT3s4 is constructed. A new algorithm of shooting technique for solving two-point BVPs for third-order ordinary differential equations (ODEs) is presented.

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Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

Chien¹,

Senu²,

Ahmadian³

et al. 2023

JST

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This study aims to propose sixth-order two-derivative improved Runge-Kutta type methods adopted with exponentially-fitting and trigonometrically-fitting techniques for integrating a special type of third-order ordinary differential equation in the form u^''' (t)=f(t,u(t),u^' (t)). The procedure of constructing order conditions comprised of a few previous steps, k-i for third-order two-derivative Runge-Kutta-type methods, has been outlined. These methods are developed through the idea of integrating initial value problems exactly with a numerical solution in the form of linear composition of the set functions e^ѡt and e^(-ѡt)for exponentially fitted and e^iѡt and e^(-iѡt) for trigonometrically-fitted with ѡ ϵ R. Parameters of two-derivative Runge-Kutta type method are adapted into principle frequency of exponential and oscillatory problems to construct the proposed methods. Error analysis of proposed methods is analysed, and the computational efficiency of proposed methods is demonstrated in numerical experiments compared to other existing numerical methods for integrating third-order ordinary differential equations with an exponential and periodic solution.

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Exponentially Fitted and Trigonometrically Fitted Explicit Modified Runge-Kutta Type Methods for Solving y′′′x=f
Nizam Ghawadri
¹
,
Norazak Senu
²
,
Fudziah Ismail
³

et al.

Cited by 3 publications

References 21 publications

Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

Direct One-Step Method for Solving Third-Order Boundary Value Problems

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

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Exponentially Fitted and Trigonometrically Fitted Explicit Modified Runge-Kutta Type Methods for Solving y′′′x=fNizam Ghawadri1, Norazak Senu2, Fudziah Ismail3 et al.

Cited by 3 publications

References 21 publications

Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

Adaptive Polynomial Method for Solving Third-Order ODE With Application in Thin Film Flow

Direct One-Step Method for Solving Third-Order Boundary Value Problems

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

Contact Info

Product

Resources

About

Exponentially Fitted and Trigonometrically Fitted Explicit Modified Runge-Kutta Type Methods for Solving y′′′x=f
Nizam Ghawadri
¹
,
Norazak Senu
²
,
Fudziah Ismail
³

et al.