A new numerical computational method for solving nonhomogeneous time-fractional nonlinear parabolic problems in unbounded domains

Document Type : Research Article

Authors

Seyed Ali Ghanizadeh

Mohammad Hossein Akrami

Department of Applied Mathematics, Faculty of Science, Yazd University, Yazd, Iran.

10.22067/ijnao.2026.96513.1777

Abstract

‎ This paper addresses the numerical solution of nonhomogeneous time-fractional nonlinear parabolic problems defined on unbounded domains‎, ‎where challenges arise due to temporal nonlocality‎, ‎nonlinear dynamics‎, ‎and the infinite spatial extent‎. ‎To overcome these difficulties‎, ‎we propose a novel computational framework that transforms the original unbounded domain problem into an equivalent formulation on a bounded domain through the construction of absorbing boundary conditions‎. ‎Unlike existing approaches that rely on Pad'e approximations‎, ‎our method directly utilizes the fundamental properties of fractional derivatives to define these boundaries more accurately‎. ‎The resulting problem is then discretized using a high-order finite difference scheme‎, ‎allowing for efficient and precise numerical implementation‎. ‎Theoretical analysis confirms that the method is stable and ensures controlled computational error‎. ‎Numerical experiments demonstrate the practical effectiveness of the proposed approach‎, ‎achieving accurate results with significantly reduced computational cost‎. ‎This method provides a reliable and efficient tool for modeling diffusion phenomena governed by time-fractional dynamics in unbounded spatial settings‎.

Keywords

Absorbing boundary conditions

High-order finite difference

Nonlinear parabolic equations‎

Subjects

• Numerical methods for differential equations and its applications

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