Recent Trends in Computational Physics with Fractional Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 404

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Department of Mathematics, Prairie View A&M University, Prairie View, TX 77446, USA
Interests: soliton theory; fractional differential equations and their applications; nonlinear dynamics and chaos; nonlinear wave theory; analytical and numerical methods for differential equations; mathematical modeling of flow in porous media
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Department of Mathematics, Nevşehir Haci Bektaş Veli Üniversitesi, Nevsehir 50300, Turkey
Interests: numerical and analytical methods for differential equations; numerical analysis; fractional differential equations and their applications
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Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USA
Interests: partial differential equations; nonlinear wave theory; integrable systems; soliton theory; Hamiltonian systems; algebro-geometric solutions; Riemann–Hilbert problem
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Department of Mathematics, West Texas A&M University, Canyon, TX 79016, USA
Interests: nonlinear wave theory; partial differential equations; dynamical systems; fractional calculus; analytical and numerical methods for differential equations; numerical methods; mathematical modeling

Special Issue Information

Dear Colleagues,

The field of computational physics has witnessed remarkable advancements over recent decades, with applications spanning a broad spectrum of scientific and engineering domains. Among these, fractional calculus has emerged as a powerful mathematical framework, extending traditional concepts of differentiation and integration to non-integer orders. This generalization has opened new avenues for modeling complex systems and processes characterized by memory effects, nonlocality, and anomalous dynamics, which are often beyond the scope of classical approaches.

Fractional calculus has proven its versatility and effectiveness across diverse areas, including fluid mechanics, viscoelastic materials, control theory, signal and image processing, financial modeling, biological systems, and nanotechnology. The integration of fractional calculus with computational methods has further expanded its applicability, enabling the simulation and analysis of intricate phenomena such as anomalous diffusion, fractional random walks, and complex dynamical systems.

Nonlinear partial differential equations (NLPDEs) remain central to many of these applications, describing fundamental physical processes across disciplines like fluid dynamics, plasma physics, solid mechanics, and quantum field theory. The inherent complexity and nonlinearity of these equations pose significant challenges for analytical solutions. Computational techniques, bolstered by methods like the inverse scattering transform, the Hirota bilinear method, Painlevé analysis, and other innovative approaches, have provided crucial tools for exploring and solving NLPDEs. The incorporation of fractional-order derivatives into these models has further enhanced their ability to capture real-world phenomena with higher accuracy and realism.

This Special Issue, titled “Recent Trends in Computational Physics with Fractional Applications”, aims to gather pioneering research that highlights the intersection of computational physics and fractional calculus. It seeks to showcase recent advancements in methodologies, novel applications, and theoretical insights that push the boundaries of this interdisciplinary field. Contributions focus on the development of new computational techniques, the exploration of fractional models in diverse scientific contexts, and the integration of fractional calculus into traditional and emerging areas of physics.

Dr. Lanre Akinyemi
Dr. Mehmet Senol
Dr. Solomon Manukure
Dr. Udoh Akpan
Guest Editors

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Keywords

  • stability analysis
  • integral equations
  • semi-analytical method
  • traveling wave solutions
  • analytical and numerical methods
  • soliton theory and its applications
  • fractional calculus and its applications
  • ordinary and partial differential equations
  • symmetry analysis and conservation laws
  • mathematical modeling of flow in porous media
  • multi-scale approaches to modeling wave phenomena
  • high-order numerical differential formulas for fractional derivatives
  • numerical and computational methods in fractional differential equations

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Published Papers (1 paper)

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Research

17 pages, 2539 KiB  
Article
Advanced Numerical Scheme for Solving Nonlinear Fractional Kuramoto–Sivashinsky Equations Using Caputo Operators
by Muhammad Nadeem and Loredana Florentina Iambor
Fractal Fract. 2025, 9(7), 418; https://doi.org/10.3390/fractalfract9070418 - 26 Jun 2025
Viewed by 186
Abstract
This work reveals an advanced numerical scheme for obtaining approximate solutions to nonlinear fractional Kuramoto–Sivashinsky (K-S) equations involving Caputo derivatives. We introduce the Sumudu transform (ST), which converts the fractional derivatives into their classical counterparts to produce a nonlinear recurrence equation. By using [...] Read more.
This work reveals an advanced numerical scheme for obtaining approximate solutions to nonlinear fractional Kuramoto–Sivashinsky (K-S) equations involving Caputo derivatives. We introduce the Sumudu transform (ST), which converts the fractional derivatives into their classical counterparts to produce a nonlinear recurrence equation. By using the homotopy perturbation method (HPM), we construct a homotopy with an embedding parameter to solve this recurrence relation. Our proposed technique is known as the Sumudu homotopy transform method (SHTM), which delivers results after fewer iterations and achieves precise outcomes with minimal computational effort. The proposed technique effectively eliminates the necessity for complex discretization or linearization, making it highly suitable for nonlinear problems. We showcase two numerical cases, along with two- and three-dimensional visualizations, to validate the accuracy and effectiveness of this technique. It also produces rapidly converging series solutions that closely align with the precise results. Full article
(This article belongs to the Special Issue Recent Trends in Computational Physics with Fractional Applications)
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