# به‌کارگیری موجک چبیشف نوع دوم در حل عددی معادلات انتگرال فردهلم خطی فازی نوع دوم

Document Type : Research Paper

Authors

گروه ریاضی کاربردی، دانشکده علوم ریاضی، دانشگاه کاشان، کاشان، ایران

10.22072/wala.2019.88855.1180

Abstract

در این مقاله، حل عددی معادلات انتگرال فردهلم فازی نوع دوم با به‌کارگیری موجک چبیشف نوع دوم را مورد بررسی قرار می‌دهیم. پس از بیان تعاریف مقدماتی مرتبط با معادلات فازی و نیز ویژگی‌های اولیه موجک چبیشف نوع دوم، فرم پارامتری معادلات انتگرال فردهلم فازی نوع دوم، که در واقع دستگاهی از معادلات انتگرال فردهلم خطی در حالت غیرفازی است را معرفی می‌نماییم. سپس با به‌کارگیری موجک چبیشف نوع دوم و به روش گالرکین، معادله انتگرال فازی را به دستگاهی از معادلات جبری خطی تبدیل می‌نماییم. نهایتا پس از حل این دستگاه، تقریبی از جواب معادله انتگرال فازی به‌دست می‌آید. با ارائه چند مثال عددی، دقت روش را مورد بررسی قرار داده و مقایسه‌ای از نتایج به‌دست آمده با نتایج ارائه شده در سایر مقالات انجام می‌دهیم.

Keywords

#### References

 S. Abbasbandy and T. Allah Viranloo, Numerical solution of fuzzy differential equation by
Runge–Kutta method, Nonlinear Stud., 11(1) (2004), 7-29.
 S. Abbasbandy, E. Babolian and M. Alavi,  Numerical method for solving linear Fredholm
fuzzy integral equations of the second kind, Chaos Solitons Fractals, 31(1) (2007), 138-146.
 G.A. Anastassiou, Fuzzy Mathematcs: Approximation Theory, Springer-Verlag, Berlin,
Heidelberge, 2010.

 G.A. Anastassiou and S.G. Gal, On a fuzzy trigonometric approximation theorem of
Weirstrass-type, J. Fuzzy Math., 9(3) (2001), 701-708.

 E. Babolian, H. Sadeghi Goghary and S. Abbasbandy, Numerical solution of linear fredholm
fuzzy integral equations of the second kind by Adomian method, Appl. Math. Comput.
161(3) (2005), 733-744.
 S. Biswas and T. Kumar Roy, Fuzzy linear integral equation and its application in
biomathematical model, Advances in Fuzzy Mathematics, 12(5) (2017), 1137-1157.

 M. Caldas and S. Jafari, \$theta\$-Compact fuzzy topological spaces, Chaos Solitons Fractals,
25(1) (2005), 229-232.

 SL. Chang and L.A. Zadeh, On fuzzy mapping and control, IEEE Trans. Syst. Man Cybern.},
2(1) (1972), 30-34.
 W. Congxin and M. Ming, On embedding problem of fuzzy number spaces, Fuzzy Sets
Syst., 44(1) (1991), 33-38.
 P. Diamond, Theory and applications of fuzzy Volterra integral equations,

IEEE Trans. Fuzzy Syst., 10(1) (2002), 97-102.
 D. Dubois and H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy
mappings, Fuzzy Sets Syst., 8(1) (1982), 1-7.
 R. Ezzati and S. Ziari, Numerical solution of nonlinear fuzzy Fredholm integral equations
using iterative method, Appl. Math. Comput., 225(1) (2013), 33-42.
 M. Friedman, M. Ma and A. Kandel, Numerical solutions of fuzzy differential and integral
equations, Fuzzy Sets Syst., 106(1) (1999), 35-48.
 S. Gal, Approximation Theory in Fuzzy Setting, In Handbook of Analytic–Computational Methods in
Applied Mathematics, Boca Raton, New York, Chapman CRC, 2000.
 M. Ghanbari, R. Toushmalni and E. Kamrani, Numerical solution of linear fredholm fuzzy
integral equation of the second kind by block-pulse functions, Australian Journal of Basic and
Applied Sciences, 3(3) (2009), 2637-2642.

 R. Goetschel and W. Vaxman, Elementary calculus, Fuzzy Sets Syst., {bf 18}(1) (1986),
31-43.
 D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications,

Philadelphia : Society for Industrial and Applied Mathematics, 1977.
 J.S. Gu and W.S. Jiang, The Haar wavelets operational matrix of integration, Int. J. Syst.
Sci., {bf 27} (1996), 623-628.
 S.M. Hashemiparast, M. Sabzevari and H. Fallahgoul, Improving the solution of nonlinear
Volterra integral equations using rationalized Haar s-functions, Vietnam J. Math., 39
(2011), 145-157.
 S.M. Hashemiparast, M. Sabzevari and H. Fallahgoul, Using crooked lines for the higher
accuracy in system of integral equations, J. Appl. Math. Comput., 29}(1-2) (2011), 145-159.

 O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24(3) (1987), 301-317.
 M. Matloka, On fuzzy integrals. In: proc 2nd polish symp on interval and fuzzy
mathematics, Wydawnicatwo Politechniki Poznanskiej, (1987), 167-170.
 F. Mirzaee, M. Komak Yari and S.F. Hoseini, A computational method based on hybrid of
Bernstein and block-pulse functions for solving linear fuzzy Fredholm integral equations
system, Journal of Taibah University for Science, 9(2) (2015), 252--263.
 A. Molabahrami, A. Shidfar and A. Ghyasi, An analytical method for solving linear Fredholm
fuzzy integral equations of the second kind, Comput. Math. Appl., 61(9) (2011), 754-2761.
 J.H. Park, Intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 22(5) (2004),
1039-1046.
 A. Saadatmandi and  M. Dehghan, A collocation method for solving Abel’s integral
equations of first and second kinds, Z. Nat.forsch., A: Phys. Sci., 63(12) (2008), 752-756.
 A. Saadatmandi and  M. Dehghan, A Legendre collocation method for fractional integro-
differential equations, J. Vib. Control, 17(13) (2011), 2050-2058.

 H. Sadeghi Goghary and M. Sadeghi Goghary, Two computational methods for solving
linear Fredholm fuzzy integral equation of the second kind, Appl. Math. Comput., 182(1)
(2006), 791-796.
 HC. Wu, On the integrals, series and integral equations of fuzzy set valued functions,

J. Harbin Inst. Technol., 21} (1990), 11-19.
 H.C. Wu, The fuzzy Riemann integral and numerical integration, Fuzzy Sets Syst., 110
(2000), 1-25.

 L.A. Zadeh, Toward a generalized theory of uncertainty (GTU)–an outline, Inf. Sci., 172
(2005), 1-40.
 L. Zhu and Q. Fan, Solving fractional nonlinear Fredholm integro-differential equations by
the second kind Chebyshev wavelet, Commun. Nonlinear Sci. Numer. Simul., 17 (2012),
2333-2341.