Dynamic response of imprecisely defined beam subject to various loads using Adomian decomposition method

Numerical solution of fuzzy uncertain beam equation using Adomian decomposition method subject to unit step and impulse loads.Fuzziness appeared in the initial conditions are modeled through convex normalized fuzzy sets viz. triangular fuzzy numbers.Adomian decomposition method (ADM) and double parametric form is used with fuzzy based approach to obtain the uncertain bounds of the dynamic responses.Obtained results are depicted in term of plots and tables to show the efficacy and powerfulness of the methodology. Present paper proposes a new technique based on double parametric form of fuzzy numbers to solve an uncertain beam equation using Adomian decomposition method subject to unit step and impulse loads. Uncertainties appear in the initial conditions are considered in terms of triangular convex normalized fuzzy sets. Using the single parametric form viz. α-cut form of fuzzy numbers, the fuzzy beam equation is converted first to an interval based fuzzy differential equation. Next this differential equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same is solved by Adomian decomposition method symbolically to obtain the uncertain bounds of the dynamic response. Obtained results are depicted in term of plots to show the efficiency and powerfulness of the present analysis.

[1]  Ji-Huan He.  A new approach to nonlinear partial differential equations , 1997 .

[2]  Esmail Babolian,et al.  Numerically solution of fuzzy differential equations by Adomian method , 2004, Appl. Math. Comput..

[3]  Barnabás Bede,et al.  Note on "Numerical solutions of fuzzy differential equations by predictor-corrector method" , 2008, Inf. Sci..

[4]  Saeid Abbasbandy,et al.  Numerical Solutions of Fuzzy Differential Equations by Taylor Method , 2002 .

[5]  Lei Wang,et al.  Adomian Method for Second-order Fuzzy Differential Equation , 2011 .

[6]  梁祖峰,et al.  Analytical solution of fractionally damped beam by Adomian decomposition method , 2007 .

[7]  S. Seikkala.  On the fuzzy initial value problem , 1987 .

[8]  Juan J. Nieto,et al.  Variation of constant formula for first order fuzzy differential equations , 2011, Fuzzy Sets Syst..

[9]  Bobby Schmidt,et al.  Fuzzy math , 2001 .

[10]  George Papageorgiou,et al.  Runge-Kutta methods for fuzzy differential equations , 2009, Appl. Math. Comput..

[11]  Y. Cherruault,et al.  New results of convergence of Adomian’s method , 1999 .

[12]  Hans-Jürgen Zimmermann,et al.  Fuzzy Set Theory - and Its Applications , 1985 .

[13]  Zhang Yue,et al.  Frequency domain methods for the solutions of N-order fuzzy differential equations , 1998, Fuzzy Sets Syst..

[14]  Saeid Abbasbandy,et al.  NUMERICAL METHODS FOR FUZZY DIFFERENTIAL INCLUSIONS , 2004 .

[15]  M. Ghanbari,et al.  NUMERICAL SOLUTION OF FUZZY INITIAL VALUE PROBLEMS UNDER GENERALIZED DIFFERENTIABILITY BY HPM , 2009 .

[16]  H. Román-Flores,et al.  On new solutions of fuzzy differential equations , 2008 .

[17]  S. Chakraverty,et al.  Approximate solution of fuzzy quadratic Riccati differential equations , 2013 .

[18]  Tofigh Allahviranloo,et al.  Nth-order fuzzy linear differential equations , 2008, Inf. Sci..

[19]  James J. Buckley,et al.  Fuzzy initial value problem for Nth-order linear differential equations , 2001, Fuzzy Sets Syst..

[20]  T. Ross.  Fuzzy Logic with Engineering Applications , 1994 .

[21]  Snehashish Chakraverty,et al.  Numerical Solution of n-th Order Fuzzy Linear Differential Equations by Homotopy Perturbation Method , 2013 .

[22]  G. Adomian.  A review of the decomposition method in applied mathematics , 1988 .

[23]  Snehashish Chakraverty,et al.  A new approach to fuzzy initial value problem by improved Euler method , 2012 .

[24]  Osmo Kaleva.  The Cauchy problem for fuzzy differential equations , 1990 .

[25]  Tofigh Allahviranloo,et al.  A method for solving nth order fuzzy linear differential equations , 2009, Int. J. Comput. Math..

[26]  D. Dubois,et al.  Towards fuzzy differential calculus part 3: Differentiation , 1982 .

[27]  Nasser Mikaeilvand,et al.  Solving fuzzy partial differential equations by fuzzy two-dimensional differential transform method , 2012, Neural Computing and Applications.

[28]  Saeid Abbasbandy,et al.  Numerical methods forfuzzy differential inclusions , 2004 .

[29]  D. Baleanu,et al.  The variational iteration method for solving n-th order fuzzy differential equations , 2012 .

[30]  P. Prakash,et al.  Numerical solution of hybrid fuzzy differential equations by predictor-corrector method , 2009, Int. J. Comput. Math..

[31]  Tapan Kumar Roy,et al.  First Order Linear Non Homogeneous Ordinary Differential Equation in Fuzzy Environment , 2013 .

[32]  Lotfi A. Zadeh,et al.  On Fuzzy Mapping and Control , 1996, IEEE Trans. Syst. Man Cybern..

[33]  Nouredin Parandin.  Numerical solution of fuzzy differential equations of nth-order by Runge–Kutta method , 2012, Neural Computing and Applications.

[34]  Abdul-Majid Wazwaz,et al.  A reliable modification of Adomian decomposition method , 1999, Appl. Math. Comput..

[35]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[36]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[37]  Bernard De Baets,et al.  Analytical and numerical solutions of fuzzy differential equations , 2013, Inf. Sci..

[38]  Mir Sajjad Hashemi,et al.  Series Solution of the System of Fuzzy Differential Equations , 2012, Adv. Fuzzy Syst..

[39]  G. Adomian.  A new approach to nonlinear partial differential equations , 1984 .

[40]  Y. Cherruault,et al.  New ideas for proving convergence of decomposition methods , 1995 .

[41]  Luc Jaulin,et al.  Applied Interval Analysis , 2001, Springer London.

[42]  George Adomian,et al.  Linear and nonlinear Schrödinger equations , 1991 .

[43]  Y. Cherruault,et al.  Convergence of Adomian's method applied to differential equations , 1994 .

[44]  T. Allahviranloo,et al.  SOLUTION OF FUZZY DIFFERENTIAL EQUATIONS UNDER GENERALIZED DIFFERENTIABILITY BY ADOMIAN DECOMPOSITION METHOD , 2009 .

[45]  I. Burhan Türksen,et al.  An algorithm for the solution of second order fuzzy initial value problems , 2013, Expert Syst. Appl..

[46]  Abraham Kandel,et al.  Numerical solutions of fuzzy differential equations , 1999, Fuzzy Sets Syst..

[47]  G. Adomian,et al.  SOLUTIONS OF NONLINEAR P.D.E. , 1998 .

[48]  S. Chakraverty,et al.  Numerical solution of fractionally damped beam by homotopy perturbation method , 2013 .

[49]  Michael Hanss,et al.  Applied Fuzzy Arithmetic: An Introduction with Engineering Applications , 2004 .

[50]  Tofigh Allahviranloo,et al.  The exact solutions of fuzzy wave-like equations with variable coefficients by a variational iteration method , 2011, Appl. Soft Comput..

[51]  Y. Cherruault.  Convergence of Adomian's method , 1989 .

[52]  M. Ghanbari.  Solution of the first order linear fuzzy differential equations by some reliable methods , 2012 .