دانشگاه تبریز
فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری
پایگاه استنادی علوم جهان اسلام (ISC)

 آمار

تعداد نشریات45
تعداد شماره‌ها1,441
تعداد مقالات17,729
تعداد مشاهده مقاله57,880,049
تعداد دریافت فایل اصل مقاله19,503,896
Nemati, Somayeh. (1405). A novel approach using a new generalization of Bernoulli wavelets for solving fractional integro-differential equations with singular kernel. سامانه مدیریت نشریات علمی, 14(1), 274-291. doi: 10.22034/cmde.2024.62229.2728
Somayeh Nemati. "A novel approach using a new generalization of Bernoulli wavelets for solving fractional integro-differential equations with singular kernel". سامانه مدیریت نشریات علمی, 14, 1, 1405, 274-291. doi: 10.22034/cmde.2024.62229.2728
Nemati, Somayeh. (1405). 'A novel approach using a new generalization of Bernoulli wavelets for solving fractional integro-differential equations with singular kernel', سامانه مدیریت نشریات علمی, 14(1), pp. 274-291. doi: 10.22034/cmde.2024.62229.2728
Nemati, Somayeh. A novel approach using a new generalization of Bernoulli wavelets for solving fractional integro-differential equations with singular kernel. سامانه مدیریت نشریات علمی, 1405; 14(1): 274-291. doi: 10.22034/cmde.2024.62229.2728

A novel approach using a new generalization of Bernoulli wavelets for solving fractional integro-differential equations with singular kernel

Computational Methods for Differential Equations
مقاله 19، دوره 14، شماره 1، فروردین 2026، صفحه 274-291    اصل مقاله (1.15 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.62229.2728
نویسنده
Somayeh Nemati*
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.
چکیده
In recent years, numerous fractional-order basis functions have been developed and applied for solving different classes of fractional problems. In this work, a new generalization of fractional-order Bernoulli wavelets is introduced. These new basis functions are used to provide a numerical solution for Hammerstein-type fractional integro-differential equations with a weakly singular kernel. To achieve this, the Riemann-Liouville integral operator is applied to the basis functions, and the result is computed exactly using the analytic form of Bernoulli polynomials. Through this process, key properties of the Riemann-Liouville integral and Caputo derivative are utilized to define two remainders associated with the main problem. After that, using an appropriate set of collocation points, the problem is converted to a system of algebraic equations. Due to the efficiency and high accuracy of this new technique, we extend the method for solving fractional Fredholm-Volterra integro-differential equations. Then, an upper bound of the error is discussed for the approximation of a function based on the fractional-order Bernoulli wavelets. Finally, the method is utilized for solving some illustrative examples to check its performance.
کلیدواژه‌ها
Weakly singular fractional integro-differential equations؛ Generalized fractional-order Bernoulli wavelets؛ Caputo derivative؛ Riemann-Liouville integral
مراجع
  • [1] D. Baleanu, R. Garra, and I. Petras, A fractional variational approach to the fractional basset-type equation, Rep. Math. Phys., 72 (2013), 57–64.
  • [2] Z. Barary, A. Yazdani Cherati, and S. Nemati, An efficient numerical scheme for solving a general class of fractional differential equations via fractional-order hybrid Jacobi functions, Commun. Nonlinear Sci. Numer. Simul., 128 (2024), 107599.
  • [3] S. Behera and S. Saha Ray, On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels, Comp. Appl. Math., 41 (2022), 211.
  • [4] J. Biazar and K. Sadri, Solution of weakly singular fractional integro-differential equations by using a new operational approach, J. Comput. Appl. Math., 352 (2019), 453–477.
  • [5] R. A. Devore and L. R. Scott, Error bounds for Gaussian quadrature and weighted L1 polynomial approximation, SIAM J. Numer. Anal., 21 (1984), 400–412.
  • [6] R. Hilfer (Ed.), Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
  • [7] P. K. Kythe and P. Puri, Computational method for linear integral equations, Birkhäuser, Boston, 2002.
  • [8] C. Li and M. Cai, Theory and numerical approximations of fractional integrals and derivatives, SIAM, 2019.
  • [9] S. Momani, Local and global existence theorems on fractional integro-differential equations, J. Fract. Calc., 18 (2000), 81–86.
  • [10] S. Momani, A. Jameel, and S. Al-Azawi, Local and global uniqueness theorems on fractional integro-differential equations via Biharis and Gronwalls inequalities, Soochow J. Math., 33 (2007), 619–627.
  • [11] S. Nemati and P. M. Lima, Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions, Appl. Math. Comput., 327 (2018), 79–92.
  • [12] S. Nemati, S. Sedaghat, and I. Mohammadi, A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels, J. Comput. Appl. Math., 308 (2016), 231–242.
  • [13] I. Podlubny, Fractional differential equations, in: Mathematics in science and engineering, Academic Press, 198 (1999).
  • [14] O. Postavaru, Generalized fractional-order hybrid of block-pulse functions and Bernoulli polynomials approach for solving fractional delay differential equations, Soft Comput., 27 (2023), 737–749.
  • [15] O. Postavaru and A. Toma, A numerical approach based on fractional-order hybrid functions of block-pulse and Bernoulli polynomials for numerical solutions of fractional optimal control problems, Math. Comput. Simul., 194 (2022), 269–284.
  • [16] P. Rahimkhani, Y. Ordokhani, and E. Babolian, Fractional-order Bernoulli wavelets and their applications, Appl. Math. Model., 40 (2016), 8087–8107.
  • [17] D. Rani and V. Mishra, Numerical inverse Laplace transform based on Bernoulli polynomials operational matrix for solving nonlinear differential equations, Results Phys., 16 (2020), 102836.
  • [18] S. Sabermahani, Y. Ordokhani, and S. A. Yousefi, Fractional-order Fibonacci-hybrid functions approach for solving fractional delay differential equations, Eng. Comput., 36 (2020), 795–806.
  • [19] S. A. Sajjadi, H. Saberi Najafi, and H. Aminikhah, A numerical study on the non-smooth solutions of the nonlinear weakly singular fractional Volterra integro-differential equations, Math. Methods Appl. Sci., 46(4) (2023), 4070– 4084.
  • [20] B. Q. Tang and X. F. Li, Solution of a class of Volterra integral equations with singular and weakly singular kernels, Appl. Math. Comput., 199 (2008), 406–413.
  • [21] Y. Wang and L. Zhu, SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput. 275 (2016), 72–80.
  • [22] M. Yi and J. Huang, CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel, Int. J. Comput. Math., 92 (2015), 1715–1728.
  • [23] M. Yi, L. Wang, and J. Huang, Legendre wavelets method for the numerical solution of fractional integrodifferential equations with weakly singular kernel, Appl. Math. Model., 40 (2016), 3422–3437.
  • [24] J. Zhao, J. Xiao, and N. J. Ford, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer. Algorithms, 65 (2014), 723–743.
  • [25] F. Zhou and X. Xu, Fractional-order hybrid functions combining simulated annealing algorithm for solving fractional pantograph differential equations, J. Comput. Sci., 74 (2023), 102172.
  • [26] V. V. Zozulya and P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-d elasticity and fracture mechanics, J. Chin. Inst. Eng., 22 (1999), 763–775.
آمار
تعداد مشاهده مقاله: 198
تعداد دریافت فایل اصل مقاله: 317


سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب