Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems

In this paper, we investigate the analytic and approximate solutions of second-order, two-point fuzzy boundary value problems based on the reproducing kernel theory under the assumption of strongly generalized differentiability. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation in the space $$\oplus _{j=1}^2 W_2^3 \left[ {a,b}\right] $$⊕j=12W23a,b. An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed method. Results of numerical experiments are provided to illustrate the theoretical statements in order to show potentiality, generality, and superiority of our algorithm for solving such fuzzy equations. Graphical results, tabulated data, and numerical comparisons are presented and discussed quantitatively to illustrate the possible fuzzy solutions.

[1]  Barnabás Bede,et al.  A note on "two-point boundary value problems associated with non-linear fuzzy differential equations" , 2006, Fuzzy Sets Syst..

[2]  Minggen Cui,et al.  Nonlinear Numerical Analysis in Reproducing Kernel Space , 2009 .

[3]  J. Nieto,et al.  A boundary value problem for second order fuzzy differential equations , 2010 .

[4]  A. Khastan,et al.  New Results on Multiple Solutions for th-Order Fuzzy Differential Equations under Generalized Differentiability , 2009 .

[5]  Mohammed Al-Smadi,et al.  Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method , 2013, Appl. Math. Comput..

[6]  V. Lakshmikantham,et al.  TWO-POINT BOUNDARY VALUE PROBLEMS ASSOCIATED WITH NON-LINEAR FUZZY DIFFERENTIAL EQUATIONS , 2001 .

[7]  R. Goetschel,et al.  Elementary fuzzy calculus , 1986 .

[8]  Mohammed Al-Smadi,et al.  Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations , 2014, Appl. Math. Comput..

[9]  Chun-li Li,et al.  The exact solution for solving a class nonlinear operator equations in the reproducing kernel space , 2003, Appl. Math. Comput..

[10]  Juan J. Nieto,et al.  NUMERICAL SOLUTION OF FUZZY DIFFERENTIAL EQUATIONS UNDER GENERALIZED DIFFERENTIABILITY , 2009 .

[11]  Omar Abu Arqub,et al.  Existence, Uniqueness, and Characterization Theorems for Nonlinear Fuzzy Integrodifferential Equations of Volterra Type , 2015 .

[12]  Zhong Chen,et al.  Solving a system of linear Volterra integral equations using the new reproducing kernel method , 2013, Appl. Math. Comput..

[13]  S. Li,et al.  A numerical method for singularly perturbed turning point problems with an interior layer , 2014, J. Comput. Appl. Math..

[14]  Sorin G. Gal,et al.  Almost periodic fuzzy-number-valued functions , 2004, Fuzzy Sets Syst..

[15]  Shaher Momani,et al.  A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Volterra type , 2014, Appl. Math. Comput..

[16]  Daniel Alpay,et al.  Reproducing Kernel Spaces and Applications , 2012 .

[17]  Shaher Momani,et al.  Analytical Solutions of Fuzzy Initial Value Problems by HAM , 2013 .

[18]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[19]  Sahin Emrah Amrahov,et al.  Solution method for a boundary value problem with fuzzy forcing function , 2015, Inf. Sci..

[20]  Omar Abu Arqub,et al.  Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations , 2017, Neural Computing and Applications.

[21]  X. Guo,et al.  Fuzzy approximate solutions of second-order fuzzy linear boundary value problems , 2013 .

[22]  Fazhan Geng,et al.  A reproducing kernel method for solving nonlocal fractional boundary value problems , 2012, Appl. Math. Lett..

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

[24]  Xiaoping Xue,et al.  Two-point boundary value problems of undamped uncertain dynamical systems , 2008, Fuzzy Sets Syst..

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

[26]  Maryam Mohammadi,et al.  Solving the generalized regularized long wave equation on the basis of a reproducing kernel space , 2011, J. Comput. Appl. Math..

[27]  Maryam Mosleh,et al.  Approximate solution of fuzzy differential equations under generalized differentiability , 2015 .

[28]  Eyke Hüllermeier,et al.  An Approach to Modelling and Simulation of Uncertain Dynamical Systems , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[29]  A. Berlinet,et al.  Reproducing kernel Hilbert spaces in probability and statistics , 2004 .

[30]  Guoqing Liu,et al.  On fuzzy boundary value problems , 2008, Inf. Sci..

[31]  Sahin Emrah Amrahov,et al.  Solution of linear differential equations with fuzzy boundary values , 2014, Fuzzy Sets Syst..

[32]  F. Z. Geng,et al.  Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers , 2013, Appl. Math. Lett..

[33]  Bo Han,et al.  A new method for solving a class of singular two-point boundary value problems , 2008, Appl. Math. Comput..

[34]  Omar Abu Arqub,et al.  The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations , 2016 .

[35]  Shaher Momani,et al.  Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method , 2016, Appl. Math. Comput..

[36]  V. Lakshmikantham,et al.  Initial and boundary value problems for fuzzy differential equations , 2003 .

[37]  P. Level,et al.  Fuzzy behavior of mechanical systems with uncertain boundary conditions , 2000 .

[38]  Wei Jiang,et al.  A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation , 2014 .

[39]  Tasawar Hayat,et al.  Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method , 2015, Soft Computing.

[40]  Xiaoping Xue,et al.  Two-point boundary value problems of uncertain dynamical systems , 2011, Fuzzy Sets Syst..