Some modifications on RCAM for getting accurate closed-form approximate solutions of Duffing- and Lienard-type equations

Das, Prakash Kumar; Singh, Debabrata; Panja, M. M.

doi:10.24297/jam.v16i0.8017

Cited by 13 publications

(5 citation statements)

References 36 publications

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“…The exact solution of the IVP (4) is u(x,t) = sin(t). Table 4: Comparison for numbers of steps between RK4 and RKF methods required to approximate u(t) at the rightmost boundary value t = 2 of the chosen interval [0, 2] for IVP (4). Per-step error tolerance of 10 E−04 is used.…”

Section: Examplementioning

confidence: 99%

“…In fact, in latter, the authors found solutions approximated by a semi-analytical method which is nothing but an intelligent combination of Differential transforms method, Laplace transform and Pade approximation. An efficient scheme based on modified Adomian decomposition method is proposed in [4] to obtain some accurate closed-form approximations of solutions to nonlinear duffing oscillator. Among numerical methods, RK4 method is considered as an effective method from the accuracy point of view, as quite often it is used as a benchmark solution.…”

Section: Examplementioning

confidence: 99%

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Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

Chowdhury

Clayton

Lemma

2019

JAM

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We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.

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Section: Examplementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

Chowdhury

Clayton

Lemma

2019

JAM

View full text Add to dashboard Cite

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“…A few analytical methods such as method based on symmetry analysis (Lie group theoretic approach) [1,2], Hirota bilinear method [3], inverse scattering transformation method [4,5], Prelle-Singer method [6], the method involving Jacobi last multiplier [7], Tanh, Sech, Exp method [8,9], Jacobi elliptic function method [10] and so on, analytical approximation schemes such as homotopy analysis method (HAM) [11,12], Adomian decomposition method (ADM) [13], Fourier transform Adomian decomposition method (FTADM) [14], rapidly convergent approximation method (RCAM) [15][16][17][18][19][20][21][22][23] etc., numerical methods viz. finite difference/element methods [24,25], Galerkin or collocation methods are used to find the solution of mathematical models.…”

Section: Introductionmentioning

confidence: 99%

A rapidly convergent approximation scheme for nonlinear autonomous and non-autonomous wave-like equations

Das¹,

Panja

2021

Filomat

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In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of exponential instead of an algebraic function of independent variables. As a consequence: i) the convergence of the series found to be faster than the same obtained by few other methods and ii) the exact analytic solution can be obtained from the first few terms of the series of the approximate solution, in cases the equation is integrable. The convergence of the sum of the successive correction terms has been established and an estimate of the error in the approximation has also been presented. The efficiency of the present method has been illustrated through some examples with a variety of nonlinear terms present in the equation.

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“…These approximate methods are often found to be slowly convergent and unable to provide the close form of the series solution. These problems can be easily tackled by the RCAM [47][48][49][50][51][52][53][54]. In this article, this scheme is used to obtain a new multi-hump travelling wave solution of a coupled Korteweg-de Vries equations with conformable derivative.…”

Section: Introductionmentioning

confidence: 99%

New multi-hump exact solitons of a coupled Korteweg-de-Vries system with conformable derivative describing shallow water waves via RCAM

Das¹

2020

Phys. Scr.

Self Cite

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In this article, a modification of the rapidly convergent approximation method is proposed to solve a coupled Korteweg–de Vries equations with conformable derivative that govern shallow-water waves. Based on the Leibniz and chain rule of conformable derivative, these equations reduced into ODEs with integer-order using traveling wave transformation. Adopting the modified scheme a new novel exact solution of the reduced coupled ordinary differential equations is obtained in terms of exponential functions. Finally, by putting them back into traveling wave transformation the solutions of the considered partial differential equations with conformable derivative are derived. To ensure the boundedness of the derived solutions few theorems have been proposed and proved. The derived results of the theorems are utilized to plot the solutions. Graphics exhibit that solutions have variant multi-hump soliton peculiarities and their tails decay to zero exponentially in a monotonic manner. These results not only show the efficiency of the modified scheme but also establish that the solution is enriched with new multi-hump features.

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Some modifications on RCAM for getting accurate closed-form approximate solutions of Duffing- and Lienard-type equations

Cited by 13 publications

References 36 publications

Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

A rapidly convergent approximation scheme for nonlinear autonomous and non-autonomous wave-like equations

New multi-hump exact solitons of a coupled Korteweg-de-Vries system with conformable derivative describing shallow water waves via RCAM

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