The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

Lu, Yanfei; Yin, Qinye; Li, Hongyi; Sun, Hongli; Hou, Muzhou

doi:10.1186/s13662-019-2131-3

Cited by 12 publications

(17 citation statements)

References 49 publications

(62 reference statements)

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“…Example Consider the following third‐order IVP 38 :

v^{(3)} (t) + v^{2} (t) = {(t - \sin t)}^{2} - \cos t; 0 \leq t \leq 1,

v (0) = v^{(1)} (0) = v^{(2)} (0) = 0,

with the exact smooth solution:

v false(t false) = - normalsin t + t

. We apply our algorithm for the case m = 3.…”

Section: Numerical Experiments and Comparisonsmentioning

confidence: 99%

“…Example Consider the following fourth‐order IVP 38 :

\begin{matrix} v^{false(4 false)} false(t false) - \frac{t^{2}}{{false(1 + v false(t false) false)}^{2}} & = - 72 t^{2} - \frac{t^{2}}{{((), 1 + {false(t - 1 false)}^{3})}^{2} t^{6} + 1} - 72 {false(1 - t false)}^{2} + 216 t false(1 - t false); 0 \leq t \leq 1, \\ v false(0 false) & = v^{false(1 false)} false(0 false) = v^{false(2 false)} false(0 false) = v^{false(3 false)} false(0 false) = 0, \end{matrix}

with the exact smooth solution:

v false(t false) = ((), {false(1 - t false)}^{3} - 1) t^{3}

. If we apply our algorithm for the case corresponding to

m = 4

and

N = 2

, then we get

c_{0} = - \frac{15}{8}, c_{1} = 1, c_{2} = - \frac{1}{8},

and consequently

v (t) = - t^{6} + 3 t^{5} - 3 t^{...}

…”

Section: Numerical Experiments and Comparisonsmentioning

confidence: 99%

See 1 more Smart Citation

A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

Youssri

Abd-Elhameed

Abdelhakem

2021

Math Methods in App Sciences

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New modified shifted Chebyshev polynomials (MSCPs) have been constructed over the interval [α, β]. These polynomials are utilized as basis functions with the application of the spectral collocation method. The operational matrix of integer order derivatives of these polynomials is introduced. The elements of this matrix are explicitly given. The introduced operational matrix along with the collocation method is used to find a direct solver of linear/nonlinear class of IVPs. Furthermore, the convergence and error analysis of the modified Chebyshev expansion are discussed. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithm. The obtained results from the tested numerical examples are convincing. Also, the introduced approximate solutions are very close to the analytical ones.

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“…Example Consider the following third‐order IVP 38 :

v^{(3)} (t) + v^{2} (t) = {(t - \sin t)}^{2} - \cos t; 0 \leq t \leq 1,

v (0) = v^{(1)} (0) = v^{(2)} (0) = 0,

with the exact smooth solution:

v false(t false) = - normalsin t + t

. We apply our algorithm for the case m = 3.…”

Section: Numerical Experiments and Comparisonsmentioning

confidence: 99%

“…Example Consider the following fourth‐order IVP 38 :

\begin{matrix} v^{false(4 false)} false(t false) - \frac{t^{2}}{{false(1 + v false(t false) false)}^{2}} & = - 72 t^{2} - \frac{t^{2}}{{((), 1 + {false(t - 1 false)}^{3})}^{2} t^{6} + 1} - 72 {false(1 - t false)}^{2} + 216 t false(1 - t false); 0 \leq t \leq 1, \\ v false(0 false) & = v^{false(1 false)} false(0 false) = v^{false(2 false)} false(0 false) = v^{false(3 false)} false(0 false) = 0, \end{matrix}

with the exact smooth solution:

v false(t false) = ((), {false(1 - t false)}^{3} - 1) t^{3}

. If we apply our algorithm for the case corresponding to

m = 4

and

N = 2

, then we get

c_{0} = - \frac{15}{8}, c_{1} = 1, c_{2} = - \frac{1}{8},

and consequently

v (t) = - t^{6} + 3 t^{5} - 3 t^{...}

…”

Section: Numerical Experiments and Comparisonsmentioning

confidence: 99%

A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

Youssri

Abd-Elhameed

Abdelhakem

2021

Math Methods in App Sciences

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“…The following example is the nonlinear ODE problem reported in [23,24], which can be summarized as follows:…”

Section: A the First Examplementioning

confidence: 99%

“…The norms of the computed errors between the exact and the estimated solutions using both the Least Square-Support Vector Machines (LS-SVMs) [23] and the proposed APM, at the discretized points of interest are tabulated in Table I.…”

Section: A the First Examplementioning

confidence: 99%

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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“…BVPs are used to model various problems in some fields, such as economics, biology, and engineering [1][2][3][4][5]. Due to the importance of ODEs, significant research work has been carried out about these problems [6][7][8][9][10][11]. In most instances, the exact solution of some ODEs cannot be obtained analytically, and numerical methods are considered as the way to obtain it.…”

Section: Introductionmentioning

confidence: 99%

Approximating Real-Life BVPs via Chebyshev Polynomials’ First Derivative Pseudo-Galerkin Method

Abdelhakem

Alaa-Eldeen²,

Băleanu

et al. 2021

Fractal Fract

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An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has been used. Consequently, new operational matrices for derivatives of any integer order have been introduced. An error analysis is performed to ensure the convergence of the presented method. In addition, the accuracy and the efficiency are verified by solving BVPs examples, including real-life problems.

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The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

Cited by 12 publications

References 49 publications

A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

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

Approximating Real-Life BVPs via Chebyshev Polynomials’ First Derivative Pseudo-Galerkin Method

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