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Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time-fractional modified Benjamin-Bona-Mahony equation

Al-deiakeh Rawaya, Alquran Marwan, Ali Mohammed, Qureshi Sania, Momani Shaher, Malkawi Abed Al-Rahman

Journal of Applied Mathematics and Computational Mechanics

2024

Vol. 23, nr 1

19-31

Lie symmetry analysis is considercd as one of the most powerful techniques that has been used for analyzing and extracting various types of solulions to partial differential equations. Conservation laws reflect important aspects of the behavior and pcoperties of physical systems. This paper focuses on the investigation of the (1+1)-dimensional time-fractional modified Benjamin-Bona-Mahony equation (mBBM) incorporating Riemann-Louville derivatives (RLD). Through the application of Lie symmetry analysis, ihe study cxplores similarity reductions and transforms the problem into a nonlinear ordinary differential equation with fractional order. A power series solution is obtained using the Erdelyi-Kober fractional operator, and the convergence of the solutions is analyzed. Furthemore, novel conservation laws for the time-fractional mBBM equation are established. The findings of the current work contribute to a deeper understanding of the dynamics of this fractional evolution equation and provide valuable insights into its behavior.

Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels

Ali Mohammed, Katatbeh Qutaibeh

Demonstratio Mathematica

2020

Vol. 53, nr 1

44--57

In this article, we study the generalized parabolic parametric Marcinkiewicz integral operators M(r)Ω,h,Φ,λ related to polynomial compound curves. Under some weak conditions on the kernels, we establish appropriate estimates of these operators. By the virtue of the obtained estimates along with an extrapolation argument, we give the boundedness of the aforementioned operators from Triebel-Lizorkin spaces to Lp spaces under weaker conditions on Ω and h. Our results represent significant improvements and natural extensions of what was known previously.