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1

On the semi-analytic technique to deal with nonlinear fractional differential equations

Khirsariya Sagar R., Rao Snehal B.

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 1

17--30

EN

In this article, we present a novel hybrid approach, by combining the Sawi transform with the homotopy perturbation method, to achieve the approximate and analytic solutions of nonlinear fractional differential equations (ODE as well as PDE) using the time-fractional Caputo derivative. The proposed algorithm is faster and simple compared to other iterative methods. The Sawi transform is used along with the homotopy perturbation method to accelerate the convergence of the series solution. The results discussed using calculations, graphs and tables are compatible for comparison with other known methods like the residual power series method and the exact solution which are discussed in the literature.

2

Solutions of Benjamin-Bona-Mahony, modified Camassa-Holm and Degasperis Procesi equations using an iterative method

Kumar Manoj

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 3

59--72

EN

In the present paper, we solve the non-linear Benjamin-Bona-Mahony, modified Camassa-Holm, and Degasperis-Procesi equations using an iterative method introduced by Daftardar-Gejji and Jafari. Results are compared with those obtained by other iterative methods such as the Adomian decomposition method and homotopy perturbation method. It is observed that the proposed method is computationally inexpensive and yields more accurate solutions than the Adomian decomposition method and the homotopy perturbation method.

3

A hybrid solution approach to the Korteweg-de Vries and Burgers' equations

Shah Kunjan, Patel Himanshu

Mathematica Applicanda

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2021

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Vol. 49, no. 2

159--170

EN

The primary purpose of this paper is to analyze the application of a new integral transform together with a homotopy perturbation method to construct approximate solutions of the initial-value problem for Korteweg-de Vries and Burgers’ equations. The new integral transform homotopy perturbation method (NIHPTM) compared to other methods, offers the simple technique to handle such type partial differential equations. The 5th-order approximation results obtained in illustrative examples compared with the explicit solutions of the considered problems show the proposed approach’s efficiency and validity.

PL

Głównym celem tego artykułu jest analiza zastosowania nowej transformacji całkowej i metody homotopijnej perturbacji do konstrukcji przybliżonych rozwiązań zagadnienia początkowego dla równań Kortewega-de Vriesa i Burgersa. Nowa metoda homotopijnej perturbacji z transformacją całkową (NIHPTM) w porównaniu z innymi metodami oferuje prostą technikę do zastosowania w tego typu równaniach różniczkowych cząstkowych. Uzyskane aproksymacje piątego rzędu dla przykładów ilustracyjnych porównane z istniejącymi jawnymi rozwiązaniami rozważanych zagadnień pokazują skuteczność i trafność proponowanego podejścia.

4

The stability conditions of the cubic damping Van der Pol-Duffing oscillator using the HPM with the frequency-expansion technology

El-Dib Y. O.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 3

31--44

EN

In this paper, we perform the frequency-expansion formula for the nonlinear cubic damping van der Pol’s equation, and the nonlinear frequency is derived. Stability conditions are performed, for the first time ever, by the nonlinear frequency technology and for the nonlinear oscillator. In terms of the van der Pol’s coefficients the stability conditions have been performed. Further, the stability conditions are performed in the case of the complex damping coefficients. Moreover, the study has been extended to include the influence of a forcing van der Pol’ oscillator. Stability conditions have been derived at each resonance case. Redoing the perturbation theory for the van der Pol oscillator illustrates more of a resonance formulation such as sub-harmonic resonance and super-harmonic resonance. More approximate nonlinear dispersion relations of quartic and quintic forms in the squaring of the extended frequency are derived, respectively.

5

Effect of variable viscosity on MHD inclined arterial blood flow with chemical reaction

Tripathi B., Sharma B. K.

International Journal of Applied Mechanics and Engineering

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2018

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Vol. 23, no. 3

767--785

EN

In this paper, we present the mathematical study of heat and mass transfer effects on an arterial blood flow under the influence of an applied magnetic field with chemical reaction. A case of mild stenosis is considered in a non-tapered artery which is inclined at an angle γ from the axis. The variable viscosity of the blood is considered varying with the hematocrit ratio. Governing non-linear differential equations have been solved by using an analytical scheme, homotopy perturbation method to obtain the solution for the velocity, temperature and concentration profiles of the blood flow. For having an adequate insight to blood flow behavior through a stenosed artery, graphs have been plotted for wall shear stress, velocity, temperature and concentration profiles with varying values of the applied magnetic field, chemical reaction parameter and porosity parameter. The results show that in an inclined artery, the magnitude of the wall shear stress at stenosis throat increases as values of the applied magnetic field increase while it reduces as the values of both the chemical reaction and porosity parameters increase. Contour plots have been plotted to show the variations of the velocity profile of blood flow as the values of the height of the stenosis as well as the influence of the applied magnetic field increase.

6

Homotopy perturbation method for solving fourth – order boundary value problems with additional boundary condition

Filipowska R.

Technical Transactions

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2017

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Vol. 9(114)

173--179

EN

This paper presents the homotopy perturbation method for solving linear and non–linear two–point boundary value problems in the form of a fourth–order differential equation and five boundary conditions. Three initial and two final conditions were taken into account. The solution of this problem is possible only when the considered equation includes an unknown parameter. The presented method has been illustrated with a numerical example.

PL

W artykule przedstawiono homotopijną metodę perturbacyjną zastosowaną do rozwiązywania zarówno liniowego, jak i nieliniowego dwupunktowego zagadnienia brzegowego składającego się z równania różniczkowego czwartego rzędu oraz pięciu warunków brzegowych. Pod uwagę wzięto trzy początkowe i dwa końcowe warunki brzegowe. Rozwiązanie tak postawionego problemu jest możliwe tylko wtedy, gdy rozpatrywane równanie zawiera nieznany parametr. Prezentowaną metodę zilustrowano przykładem obliczeniowym.

7

Application of the homotopy perturbation method for the systems of Fredholm integral equations

Hetmaniok E., Słota D., Wróbel A., Zielonka A.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

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2015

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z. 5

61--70

EN

In this paper the convergence of homotopy perturbation method for the systems of Fredholm integral equations of the second kind is proved. Estimation of errors of approximate solutions obtained by taking the partial sum of the series is also elaborated in the paper.

PL

W artykule wykazano zbieżność homotopijnej metody perturbacyjnej dla układów równań całkowych Fredholma drugiego rodzaju. Podano także oszacowanie błędu rozwiązania przybliżonego uzyskanego jako suma częściowa tworzonego w metodzie szeregu.

8

Application of the homotopy perturbation method for the systems of Volterra integral equations

Hetmaniok E., Słota D., Wróbel A., Zielonka A.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

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2015

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z. 5

71--77

EN

In this paper the convergence of homotopy perturbation method for the systems of Volterra integral equations of the second kind is proved. Estimation of errors of approximate solutions obtained by taking the partial sum of the series is also elaborated in the paper.

PL

W artykule wykazano zbieżność homotopijnej metody perturbacyjnej dla układów równań całkowych Volterry drugiego rodzaju. Podano także oszacowanie błędu rozwiązania przybliżonego uzyskanego jako suma częściowa tworzonego w metodzie szeregu.

9

Homotopy perturbation method combined with Trefftz method in numerical identification of liquid temperature in flow boiling

Hożejowska S.

Journal of Theoretical and Applied Mechanics

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2015

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Vol. 53 nr 4

969--980

EN

The paper is focused on numerical identification of 2D temperature fields in flow boiling of the liquid through a horizontal minichannel with a rectangular cross-section. The heat transfer process in the minichannel is described by a two-dimensional energy equation with the corresponding boundary conditions. Liquid temperature is determined using the homotopy perturbation method (HPM) with Trefftz functions for Laplace’a equation. The numerical solution to the energy equation found with the HPM is compared with the solution obtained for the simplified form of the energy equation. Considering that only the thermal sublayer is taken into account, both solutions give similar results.

10

Thermal performance of porous fins with temperature-dependent heat generation via the homotopy perturbation method and collocation method

Hoshyar H. A., Rahimipetroudi I., Ganji D. D., Majidian A. R.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 4

53--65

EN

An analysis has been performed to study the problem of the thermal performance of a nonlinear problem of the porous fin with temperature-dependent internal heat generation. Highly accurate semi-analytical methods called the collocation method (CM) and the homotopy perturbation method (HPM) are introduced and then are used to obtain a nonlinear temperature distribution equation in a longitudinal porous fin. This study is performed using passage velocity from the Darcy’s model to formulate the heat transfer equation through porous media. The heat generation is assumed to be a function of temperature. The effects of the natural convection parameter Nc, internal heat generation εg, porosity Sh and generation number G parameter on the dimensionless temperature distribution are discussed. Also, numerical calculations called the fourth order Runge-Kutta method were carried out for the various parameters entering into the problem for validation. Results reveal that analytical approaches are very effective and convenient. Also it is found that these methods can achieve more suitable results compared to numerical methods.

11

Non-probabilistic Solutions of Uncertain Fractional Order Diffusion Equations

Tapaswini S., Chakraverty S.

Fundamenta Informaticae

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2014

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Vol. 133, nr 1

19--34

EN

This paper investigates the numerical solution of uncertain fractional order diffusion equation subject to various external forces. Homotopy Perturbation Method (HPM) is used for the analysis. Uncertainties present in the system are modelled through triangular convex normalised fuzzy sets. A new computational technique has been proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy fractional diffusion equation is converted first to an interval fuzzy fractional differential equation. Next this equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same is solved by HPM symbolically to obtain the uncertain bounds of the solution. Obtained results are depicted in term of plots. Results obtained by the proposed method are compared with existing results in special cases.

12

An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit

Hetmaniok E., Słota D., Trawiński T., Wituła R.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2014

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Vol. 62, nr 3

413--421

EN

In this paper an application of the homotopy perturbation method for solving the general linear integral equations of the second kind is discussed. It is shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in the homotopy perturbation method is convergent. The error of approximate solution, received by taking only the partial sum of the series, is also estimated. Moreover, there is presented an example of applying the method for approximate solution of an equation which has a practical application for charge calculation in supply circuit of the flash lamps used in cameras.

13

Comparison between HPM and finite Fourier solution in static analysis of FGPM beam under thermal load

Armin A., Shafieenejad I., Moallemi N., Novinzadeh A. B.

Journal of Theoretical and Applied Mechanics

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2010

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Vol. 48 nr 1

173-189

EN

Linear and nonlinear phenomena play important role in applied mathematics, physics and also in engineering problems in which any parameter may vary depending on different factors. In recent years, the homotopy perturbation method (HPM) is constantly being developed and applied to solve various linear and nonlinear problems. In this paper, static analysis of functionally graded piezoelectric beams based on the first-order shear deformation theory under thermal loads has been investigated. The beam with a functionally graded piezoelectric material (FGPM) is graded in the thickness direction and a simple power law index governs the piezoelectric material properties. The electric potential is assumed linear across the beam thickness. The governing equations are obtained using potential energy and Hamilton's principle and may lead to a system of differential equations. We suggest two methods to solve this problem, the homotopy perturbation and analytical solution obtained by the finite Fourier transformation. The homotopy perturbation method and a proper algorithm are suggested to solve simultaneous differential equations. The results are presented for different power law indexes under uniform thermal gradient. The results are compared with the analytical solution obtained by the finite Fourier transformation for simply supported boundary conditions.

PL

Zjawiska liniowe i nieliniowe odgrywają ważna rolę w dziedzinie matematyki stosowanej, fizyki, a tak ze zagadnieniach inżynierskich, w których dowolny parametr może ulegać zmianie pod wpływem różnych czynników. W ostatnich latach perturbacyjna metoda homotopii (HPM) ulegała ciągłemu rozwojowi i znalazła zastosowanie w rozwiązywaniu różnorodnych liniowych i nieliniowych zadań. W tej pracy zaprezentowano wyniki analizy statycznej belki wykonanej z gradientowego materiału zawierającego frakcję piezoelektryczną i obciążonej termicznie otrzymanych przy pomocy teorii odkształceń postaciowych pierwszego rzędu. Belka z materiału funkcjonalnego (FGPM) ma strukturę gradientową, tj. posiada właściwości materiałowe zmienne w sposób ciągły wzdłuż grubości tej belki, zgodnie z założonym rozkładem wykładniczym zawartości aktywnej frakcji piezoelektryka w całym materiale. Założono, że potencjał elektryczny ma rozkład liniowy wzdłuż grubości belki. Różniczkowe równania ruchu układu otrzymano, używając wyrażenia na energię potencjalną i stosując zasadę Hamiltona. Do ich rozwiązania zaproponowano dwie metody: perturbacyjną homotopii i analityczną w drodze skończonej transformacji Fouriera. W metodzie homotopii zasugerowano odpowiedni algorytm rozwiązywania układu równań różniczkowych. Wyniki przedstawiono dla różnych rozkładów aktywnej frakcji piezoelektrycznej przy utrzymaniu jednorodnego gradientu temperatury. Wyniki porównano z rozwiązaniem analitycznym otrzymanym za pomocą skończonej transformacji Fouriera dla warunków brzegowych belki odpowiadających swobodnemu podparciu.

14

Application of the homotopy perturbation method for calculation of the temperature distribution in the cast-mould heterogeneous domain

Grzymkowski R., Hetmaniok E., Słota D.

Journal of Achievements in Materials and Manufacturing Engineering

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2010

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Vol. 43, nr 1

299--309

EN

Purpose of this paper: In this paper an application of the new method for solving the heat conduction equation in the heterogeneous cast-mould system, with an assumption of the ideal contact at the cast-mould contact point, is introduced. An example illustrating the discussed approach and confirming its usefulness for solving problems of that kind is also presented in the paper. Design/methodology/approach: For solving the discussed problem the homotopy perturbation method is used, which consists in determining the series convergent to the exact solution or enabling to built the approximate solution of the problem. Findings: The paper shows that the homotopy perturbation method, effective in solving many technical problems, is successful also for examining the considered problem. Research limitations/implications: Solution of the problem is provided with the assumption of an ideal contact between the cast and the mould. In further, research of the discussed method shall be employed to solve problems involving the presence of thermal resistance at the cast-mould contact Practical implications: The method allows to determine the solution in form of the continuous function, which is significant for the analysis of the cast cooling in the mould, in order to avoid the defects formation in the cast. Originality/value: Application of the new method for solving the considered problem.