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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
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