Numeric-Analytic Solutions for Nonlinear Oscillators via the Modified Multi-Stage Decomposition Method

Az-Zo’bi, Emad A.; Al‐Khaled, Kamel; Darweesh, Amer

doi:10.3390/math7060550

Cited by 28 publications

(16 citation statements)

References 36 publications

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“…From the initial value problem (1), N(u) = u 2 u and hence by using (10), the Adomian polynomials are obtained as…”

Section: Thus We Obtain the Relation For Generating The Adomian Polynmentioning

confidence: 99%

“…The van der Pol equation, otherwise called the van der Pol oscillator is a model which describes the behaviour of electrical circuits [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. It was formulated by an electrical engineer and physicist, Balthasar van der Pol, and no exact solution has been obtained for the second order differential equation since then [12].…”

Section: Introductionmentioning

confidence: 99%

“…This procedure, which generates series solutions to differential equations that converge very rapidly, is called Adomian decomposition method, named after the author [18]. The Adomian decomposition method has been used to obtain approximate analytic solutions to a wide range of differential equations, including linear and nonlinear equations, and has been shown to converge more rapidly with proofs of convergence than other forms of series solutions ( [1,4,13,17,19]). There have been various modifications to this procedure which have significantly improved the convergence of the method ( [1-3, 5, 8, 9, 14, 18]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Comparison of the Solution of the Van der Pol Equation Using the Modified Adomian Decomposition Method and Truncated Taylor Series Method

Ndam

Adedire

2020

J. Nig. Soc. Phys. Sci.

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In this paper, we compare the solution of the van der Pol equation obtained by using the truncated Taylor series method and the modified Adomian decomposition method with the solution obtained by the Poincare-Lindstedt (P-L) method. The approximating 4-component modified Adomian decomposition method behaves more like an approximate P-L analytic method than the tenth-order Taylor series. Also, with the addition of just one term, the approximating 5-component modified Adomian decomposition method produces a more convergent solution to that of P-L analytic method than the twenty second-order Taylor series approximation as the independent variable t representing time progressively increases. A general comparison of the two solutions revealed that the absolute errors generated by the approximating polynomial from the Taylor series are greater than the ones generated from the modified Adomian decomposition method. It was further revealed that very few components of the modified Adomian decomposition could yield a series of about 3 times the order of the one obtained by using the Taylor series method. Hence, we recommend the inclusion of the modified Adomian Decomposition Method in modern mathematical tools.

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“…From the initial value problem (1), N(u) = u 2 u and hence by using (10), the Adomian polynomials are obtained as…”

Section: Thus We Obtain the Relation For Generating The Adomian Polynmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Comparison of the Solution of the Van der Pol Equation Using the Modified Adomian Decomposition Method and Truncated Taylor Series Method

Ndam

Adedire

2020

J. Nig. Soc. Phys. Sci.

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show abstract

“…In recent years, researchers have been paying much attention to nonlinear problems since many practical systems are known for their nonlinear characteristics 1,2 . Many numerical methods are applied to nonlinear problems, such as the multiple‐scale method, 3 fourth‐order Runge–Kutta (RK4) method, 4 ordinary perturbation method, 5 harmonic balance (HB) method, 6 and so on.…”

Section: Introductionmentioning

confidence: 99%

Numeric‐analytical solution of an LC circuit with nonlinear capacitor based on the symplectic conservative perturbation method

Yang

Wang

et al. 2021

Int J Numerical Modelling

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The LC circuit with a nonlinear capacitor is analyzed using the symplectic conservative perturbation method based on the symplectic matrix. By treating the electric charge and the flux linkage as dual variables, the LC circuit system can be transformed into a Hamiltonian system. Then we solve the nonlinear Hamiltonian matrix equation by the perturbation method. The Hamiltonian matrix can be cast into linear and nonlinear parts. Based on the exact solution of the linear part, the nonlinear part is solved perturbatively by a canonical transformation. Since the coefficient matrix of the obtained equation is still a Hamiltonian matrix, symplectic conservation is guaranteed. The validity of the method is demonstrated, and the stability, precision, and the efficiency of the method are discussed by comparing it with the fourth‐order Runge–Kutta method (RK4), the ordinary perturbation method, the small‐parameter perturbation method based on displacement (SPPD), and the harmonic balance (HB) method. Numerical results show that the proposed method has better stability, precision, and efficiency compared with the other methods and thus has great advantages in long‐term simulations.

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“…In this direction, the literature reports on various methods to approximate the solutions of fractional systems. For example, some numerical methods have been proposed to solve fractional partial differential equations using fractional centered differences [35][36][37], the time-fractional diffusion equation [38], the fractional Schrödinger equation in multiple spatial dimensions [39], the nonlinear fractional Korteweg-de Vries-Burgers equation [40], and the fractional FitzHugh-Nagumo monodomain models [41], among other examples [42][43][44][45].…”

Section: Introductionmentioning

confidence: 99%

Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme

Macías‐Díaz

2019

Mathematics

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In this work, we investigate numerically a one-dimensional wave equation in generalized form. The system considers the presence of constant damping and functional anomalous diffusion of the Riesz type. Reaction terms are also considered, in such way that the mathematical model can be presented in variational form when damping is not present. As opposed to previous efforts available in the literature, the reaction terms are not only functions of the solution. Instead, we consider the presence of smooth functions that depend on fractional derivatives of the solution function. Using a finite-difference approach, we propose a numerical scheme to approximate the solutions of the fractional wave equation. Along with this integrator, we propose discrete forms of the local and the total energy operators. In a first stage, we show rigorously that the energy properties of the continuous system are mimicked by our discrete methodology. In particular, we prove that the discrete system is dissipative (respectively, conservative) when damping is present (respectively, absent), in agreement with the continuous model. The theoretical numerical analysis of this system is more complicated in light of the presence of the functional form of the anomalous diffusion. To solve this problem, some novel technical lemmas are proved and used to establish the stability and the quadratic convergence of the scheme. Finally, we provide some computer simulations to show the capability of the scheme to conserve/dissipate the energy. Various fractional problems with functional forms of the anomalous diffusion of the solution are considered to that effect.

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Numeric-Analytic Solutions for Nonlinear Oscillators via the Modified Multi-Stage Decomposition Method

Cited by 28 publications

References 36 publications

Comparison of the Solution of the Van der Pol Equation Using the Modified Adomian Decomposition Method and Truncated Taylor Series Method

Comparison of the Solution of the Van der Pol Equation Using the Modified Adomian Decomposition Method and Truncated Taylor Series Method

Numeric‐analytical solution of an LC circuit with nonlinear capacitor based on the symplectic conservative perturbation method

Numerically Efficient Methods for Variational Fractional Wave Equations: An Explicit Four-Step Scheme

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