Finding optimal convergence control parameter in the homotopy analysis method to solve integral equations based on the stochastic arithmetic

The goal of this paper is to present a new scheme based on the stochastic arithmetic (SA) to find the optimal convergence control parameter, the optimal iteration and the optimal approximation of the homotopy analysis method (HAM). This scheme is called the CESTAC1 method. Also, the CADNA2 library is applied to implement the CESTAC method on the proposed algorithms. CADNA is able to present a new, robust and valid environment to implement the HAM and optimize the results. By using this method, not only the optimal auxiliary control parameter can be computed but also the unnecessary iterations can be neglected and optimal step of this method is achieved. The main theorems are presented to guarantee the validity and accuracy of the HAM. Different kinds of integral equations such as singular and first kind are considered to find the optimal results by applying the proposed algorithms. The numerical results show the importance and efficiency of the SA in comparison with the floating-point arithmetic (FPA).

[1]  Fabrizio Lillo,et al.  Discrete Homotopy Analysis for Optimal Trading Execution with Nonlinear Transient Market Impact , 2016, Commun. Nonlinear Sci. Numer. Simul..

[2]  M. A. Araghi,et al.  Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method , 2017 .

[3]  Sohrab Ali Yousefi,et al.  Numerical solution of Abel's integral equation by using Legendre wavelets , 2006, Appl. Math. Comput..

[4]  Antonio Mastroberardino,et al.  Homotopy analysis method applied to electrohydrodynamic flow , 2011 .

[5]  Habibolla Latifizadeh.  APPLICATION OF HOMOTOPY ANALYSIS TRANSFORM METHOD TO FRACTIONAL BIOLOGICAL POPULATION MODEL , 2013 .

[6]  Sehar Iqbal,et al.  Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind , 2012, Appl. Math. Comput..

[7]  Magdy A. El-Tawil,et al.  A new technique of using homotopy analysis method for second order nonlinear differential equations , 2012, Appl. Math. Comput..

[8]  Faezeh Toutounian Mashhad.  Numerical accuracy of a certain class of iterative methods for solving linear system , 2006 .

[9]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[10]  Zakieh Avazzadeh,et al.  Fractional Calculus for Solving Abel's Integral Equations Using Chebyshev Polynomials , 2011 .

[11]  Yong Huang,et al.  Approximate solution of Abel integral equation , 2008, Comput. Math. Appl..

[12]  Fabienne Jézéquel,et al.  Dynamical Control of Computations Using the Trapezoidal and Simpson's Rules , 1998, J. Univers. Comput. Sci..

[13]  Saeid Abbasbandy,et al.  Mathematical properties of ℏ-curve in the frame work of the homotopy analysis method , 2011 .

[14]  J. Vignes,et al.  Validation of results of collocation methods for ODEs with the CADNA library , 1996 .

[15]  Ranjan Das,et al.  Application of homotopy analysis method and inverse solution of a rectangular wet fin , 2014 .

[16]  Robert A. Van Gorder,et al.  Optimal homotopy analysis and control of error for solutions to the non-local Whitham equation , 2013, Numerical Algorithms.

[17]  Robert A. Van Gorder,et al.  Control of error in the homotopy analysis of semi-linear elliptic boundary value problems , 2012, Numerical Algorithms.

[18]  Hasan Barzegar Kelishami,et al.  Homotopy analysis transform method for solving Abel’s integral equations of the first kind , 2016 .

[19]  O. Martin.  On the homotopy analysis method for solving a particle transport equation , 2013 .

[20]  Devendra Kumar,et al.  New homotopy analysis transform algorithm to solve volterra integral equation , 2014 .

[21]  Khaled Omrani,et al.  The homotopy analysis method for solving the Fornberg-Whitham equation and comparison with Adomian's decomposition method , 2010, Comput. Math. Appl..

[22]  F. A. Chandio,et al.  ELZAKI PROJECTED DIFFERENTIAL TRANSFORM METHOD FOR FRACTIONAL ORDER SYSTEM OF LINEAR AND NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATION , 2018 .

[23]  Mohammad Mehdi Rashidi,et al.  The homotopy analysis method for explicit analytical solutions of Jaulent–Miodek equations , 2009 .

[24]  R. N. Desmarais,et al.  Tables of properties of airfoil polynomials , 1995 .

[25]  Huazhong Shu,et al.  Numerical solutions of partial differential equations by discrete homotopy analysis method , 2010, Appl. Math. Comput..

[26]  M. A. Araghi,et al.  A novel technique based on the homotopy analysis method to solve the first kind Cauchy integral equations arising in the theory of airfoils , 2016 .

[27]  Qi Wang.  The optimal homotopy-analysis method for Kawahara equation , 2011 .

[28]  S. Abbasbandy,et al.  A stochastic scheme for solving definite integrals , 2005 .

[29]  Saeid Abbasbandy,et al.  A new analytical technique to solve Volterra's integral equations , 2011 .

[30]  Fevzi Erdogan,et al.  The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian's decomposition method , 2013 .

[31]  A. Wazwaz.  Nonlinear Singular Integral Equations , 2011 .

[32]  S. Abbasbandy.  Homotopy analysis method for heat radiation equations , 2007 .

[33]  S. Abbasbandy.  Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method , 2008 .

[34]  Zaid M. Odibat,et al.  A study on the convergence of homotopy analysis method , 2010, Appl. Math. Comput..

[35]  S. Abbasbandy,et al.  The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree , 2004 .

[36]  A. Wazwaz.  Linear and Nonlinear Integral Equations: Methods and Applications , 2011 .

[37]  M. A. Araghi,et al.  Homotopy analysis transform method for solving generalized Abel's fuzzy integral equations of the first kind , 2015, 2015 4th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS).

[38]  M. A. Araghi,et al.  Fibonacci-regularization method for solving Cauchy integral equations of the first kind , 2017 .

[39]  Jean Marie Chesneaux.  The equality relations in scientific computing , 2005, Numerical Algorithms.

[40]  S. Liao.  An optimal homotopy-analysis approach for strongly nonlinear differential equations , 2010 .

[41]  Faezeh Toutounian,et al.  The use of the CADNA library for validating the numerical results of the hybrid GMRES algorithm , 1997 .

[42]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[43]  F. Awawdeh,et al.  A Numerical Method for Solving Linear Integral Equations , 2009 .

[44]  Saeid Abbasbandy,et al.  Determination of optimal convergence-control parameter value in homotopy analysis method , 2013, Numerical Algorithms.

[45]  Saeed Kazem,et al.  A modification of the homotopy analysis method based on Chebyshev operational matrices , 2013, Math. Comput. Model..

[46]  Shijun Liao,et al.  Homotopy Analysis Method in Nonlinear Differential Equations , 2012 .

[48]  A. K. Shukla,et al.  Homotopy analysis method with a non-homogeneous term in the auxiliary linear operator , 2012 .

[49]  K. Vajravelu,et al.  On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach , 2009 .

[50]  O. Abdulaziz,et al.  Homotopy analysis method for fully developed MHD micropolar fluid flow between vertical porous plates , 2009 .

[51]  Samad Noeiaghdam,et al.  A Valid Scheme to Evaluate Fuzzy Definite Integrals by Applying the CADNA Library , 2017, Int. J. Fuzzy Syst. Appl..

[52]  Chun Wang,et al.  A one-step optimal homotopy analysis method for nonlinear differential equations , 2010 .

[53]  J. Faye,et al.  Stochastic approach of the permutation-perturbation method for round-off error analysis , 1985 .

[54]  S. Abbasbandy,et al.  Homotopy analysis method for quadratic Riccati differential equation , 2008 .

[55]  Sehar Iqbal,et al.  Numerical solutions of weakly singular Volterra integral equations using the optimal homotopy asymptotic method , 2012, Comput. Math. Appl..

[56]  A. Golbabai,et al.  Application of the optimal homotopy asymptotic method for solving a strongly nonlinear oscillatory system , 2013, Math. Comput. Model..

[57]  R. Nazar,et al.  Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method , 2009 .

[59]  Damian Slota,et al.  Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind , 2013, Numerical Algorithms.

[60]  Soheil Soleimani,et al.  Application of homotopy analysis method to solve MHD Jeffery-Hamel flows in non-parallel walls , 2011, Adv. Eng. Softw..

[61]  M. A. Araghi.  HOMOTOPY REGULARIZATION METHOD TO SOLVE THE SINGULAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND , 2018 .

[63]  Y. Tan,et al.  A General Approach to Obtain Series Solutions of Nonlinear Differential Equations , 2007 .

[64]  Mohd. Salmi Md. Noorani,et al.  Homotopy analysis method for singular IVPs of Emden–Fowler type , 2009 .

[65]  Abdelaziz Mennouni,et al.  Solving Cauchy integral equations of the first kind by the Adomian decomposition method , 2013, Appl. Math. Comput..

[66]  Jean Vignes,et al.  A stochastic arithmetic for reliable scientific computation , 1993 .

[67]  Sohrab Effati,et al.  A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems , 2013 .

[68]  Ugur Yücel,et al.  Homotopy analysis method for the sine-Gordon equation with initial conditions , 2008, Appl. Math. Comput..