Rational Homotopy Perturbation Method for solving stiff systems of ordinary differential equations

Biazar, Jafar; Asadi, Maryam; Salehi, Farideh

doi:10.1016/j.apm.2014.09.003

Cited by 24 publications

(14 citation statements)

References 24 publications

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“…It is very difficult to solve nonlinear problems and in general it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, such as, Perturbation Method [1], Homotopy Perturbation Method [2], Modified Homotopy Perturbation Method [3,4], Rational Homotopy Perturbation Method [5], He's Homotopy Perturbation Method [6], Modified He's homotopy Perturbation Method [7], Asymptotic Method [8][9][10][11], Optimal Iteration Perturbation Method [12], Generalization of Modified Differential Transforms Method [13][14][15][16], and so on. Several other authors used many powerful analytical methods in the field of approximate solutions especially for strongly nonlinear oscillators like Max-Min Approach Method [17,18], Algebraic Method [19], Parameter Expansion Method and Variational Iteration Method [20][21][22], Amplitude Frequency Formulation Method [23], Energy Balance Method [24,25], He's Energy Balance Method [26,27], Rational Energy Balance Method [28], Rational Harmonic Balance Method [29], Residue Harmonic Balance Method [30][31][32][33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs.…”

Section: Introductionmentioning

confidence: 99%

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Hosen

Chowdhury

2015

Results in Physics

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a b s t r a c tIn the present paper, a complicated strongly nonlinear oscillator with cubic and harmonic restoring force, has been analysed and solved completely by harmonic balance method (HBM). Investigating analytically such kinds of oscillator is very difficult task and cumbersome. In this study, the offered technique gives desired results and to avoid numerical complexity. An excellent agreement was found between approximate and numerical solutions, which prove that HBM is very efficient and produces high accuracy results. It is remarkably important that, second-order approximate results are almost same with exact solutions. The advantage of this method is its simple procedure and applicable for many other oscillatory problems arising in science and engineering. IntroductionNonlinear oscillations are important fact in physical science, mechanical structures and other engineering problems. Naturally, all differential equations involving engineering and physical phenomena are nonlinear. The methods of solutions of linear differential equations are comparatively easy and well established. On the contrary, the techniques of solutions of nonlinear differential equations (NDEs) are less available and have no exact solution and, in general, linear approximations are frequently used. Nowadays, NDEs have been the subject of all-embracing studies in various branches of nonlinear science and engineering. A special class of analytical solutions named strongly nonlinear oscillator with cubic and harmonic restoring force has a lot of importance, because, most of the phenomena that arise in mathematical physics and engineering fields can be described by NDEs. Therefore, investigating strongly nonlinear oscillator with cubic and harmonic restoring force solutions is becoming increasingly attractive in nonlinear sciences. Moreover, obtaining exact solutions for nonlinear oscillatory problems has many difficulties. It is very difficult to solve nonlinear problems and in general it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, of Modified Differential Transforms Method [13][14][15][16], and so on. Several other authors used many powerful analytical methods in the field of approximate solutions especially for strongly nonlinear oscillators like Max-Min Approach Method [17,18], Algebraic Method [30-33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs. The HBM is another technique for solving strongly nonlinear systems. Borges et al. [35] and Bobylev et al. [36] first provided overviews of HBM. Mickens [37-39] was first applied HBM in truly nonlinear oscillators. Due to his contribution he is known as father of HBM. Afterwards, Belendez et al. [40] and others researchers [41-43] has significantly improved the HBM.The HBM provides a general technique for calculating approximations to the periodic solutio...

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Section: Introductionmentioning

confidence: 99%

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Hosen

Chowdhury

2015

Results in Physics

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“…Consider the following nonlinear stiff systems of ODEs [38]. In this numerical example, two scheme are compared and as explained the main task of the modified method tackle the singularity of the moment matrix.…”

Section: Volterra Typementioning

confidence: 99%

Modified moving least squares method for two-dimensional linear and nonlinear systems of integral equations

Matinfar

Pourabd

2018

Comp. Appl. Math.

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This work aims at focusing on modifying the moving least squares (MMLS) methods for solving two-dimensional linear and nonlinear systems of integral equations and system of differential equations. The modified shape function is our main aim, so for computing the shape function based on the moving least squares method (MLS), an efficient algorithm is presented. In this modification, additional terms is proposed to impose based on the coefficients of the polynomial base functions on the quadratic base functions (m = 2). So, the MMLS method is developed for solving the systems of two-dimensional linear and nonlinear integral equations at irregularly distributed nodes. This approach prevents the singular moment matrix in the context of MLS based on meshfree methods. Also, determining the best radius of the support domain of a field node is an open problem for MLS-based methods. Therefore, the next important thing is that the MMLS algorithm can automatically find the best neighborhood radius for each node. Then, numerical examples are presented that determine the main motivators for doing this so. These examples enable us to make comparisons of two methods: MMLS and classical MLS.

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“…Then stiff systems were studied by many researchers in different branches [2]- [4]. Several numerical methods have been developed to obtain the analytical and approximate solutions of stiff systems of integer and fractional orders, such as Adomian decomposition method [5], homotopy perturbation method [6], homotopy analysis method [7], variation iteration method [8], rational homotopy perturbation method [9], Block method [10], Laplace adomian decomposition method and modified decomposition method [11], multistage Bernstein method [12], and fractional power series method [13].…”

Section: Introductionmentioning

confidence: 99%

A New Computational Method Based on Integral Transform for Solving Linear and Nonlinear Fractional Systems

et al. 2021

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In this article, the Elzaki homotopy perturbation method is applied to solve fractional stiff systems. The Elzaki homotopy perturbation method (EHPM) is a combination of a modified Laplace integral transform called the Elzaki transform and the homotopy perturbation method. The proposed method is applied for some examples of linear and nonlinear fractional stiff systems. The results obtained by the current method were compared with the results obtained by the kernel Hilbert space KHSM method. The obtained result reveals that the Elzaki homotopy perturbation method is an effective and accurate technique to solve the systems of differential equations of fractional order.

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Rational Homotopy Perturbation Method for solving stiff systems of ordinary differential equations

Cited by 24 publications

References 24 publications

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Modified moving least squares method for two-dimensional linear and nonlinear systems of integral equations

A New Computational Method Based on Integral Transform for Solving Linear and Nonlinear Fractional Systems

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