Research Article
Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type
Yong-Dok Han*,
Kang-Song Yun,
Chol-Gwang Kim
Issue:
Volume 10, Issue 1, June 2026
Pages:
1-13
Received:
28 January 2025
Accepted:
12 August 2025
Published:
23 January 2026
Abstract: In the mathematical theory of nerve impulse propagation, the Fitzhugh-Nagumo Reaction-Diffusion System has attracted a great deal of attention. The Fitzhugh-Nagumo Reaction-Diffusion System provides a prototype for chemical and other nerve conduction and biological systems. In this paper, we define two types of weak solutions of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System, namely (1) -weak solutions and (2) -weak solutions, and demonstrate the existence and uniqueness of these weak solutions. First, we have obtained a generalization of [1, Lemma 1] in Lemma 2.1 and using Lemma 2.1 and Galerkin’s approximation sequence, we have found the existence of (1)-weak solutions and (2)-weak solutions. We also obtained a generalization of the result of [10, Lemma 6] to Hilbert spaces in Lemma 2.2, and using this result we proved the uniqueness of the (2)-weak solution. Lemma 2.1 and Lemma 2.2 of this paper are results that can be effectively used to show the existence and uniqueness of weak solutions of time fractional partial differential equations. And the existence and uniqueness results of the weak solution of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System can be used in the numerical solutions of this reaction-diffusion system. Also, we can be used in the optimal control problems described in this system.
Abstract: In the mathematical theory of nerve impulse propagation, the Fitzhugh-Nagumo Reaction-Diffusion System has attracted a great deal of attention. The Fitzhugh-Nagumo Reaction-Diffusion System provides a prototype for chemical and other nerve conduction and biological systems. In this paper, we define two types of weak solutions of the time fractional...
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Research Article
Insight into Reformulation of an A-stable Runge-Kutta (RK) Method as an Implicit Runge-Kutta-Nystrom (RKN) Method for Directly Solving Second-order ODEs
Momoh Adelegan Lukuman*
Issue:
Volume 10, Issue 1, June 2026
Pages:
14-23
Received:
18 February 2026
Accepted:
5 March 2026
Published:
8 April 2026
DOI:
10.11648/j.engmath.20261001.12
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Abstract: This work intends to contribute to the growing literature on the direct numerical solution of special and general second-order initial value problems (IVPs) of ordinary differential equations (ODEs) by reformulating an A-stable Runge-Kutta (RK) method as a type of implicit Runge–Kutta–Nystrӧm methods (RKN). Unlike traditional methods that reduce second-order IVPs of ODEs to equivalent systems of first-order IVPs, the approach in this work preserves the original problem structure while also benefiting from the sixth-order accuracy and A-stability of the GLRK method. Using an extended Butcher array, the resulting RKN coefficients are obtained explicitly, ensuring they meet the consistency and order conditions. The linear stability analysis reveals a broad stability region, which is necessary for handling stiff and oscillatory systems. Application of the method to numerical examples shows that it offers better accuracy than most existing methods, while maintaining a similar computational cost. This reformulation strategy paves the way for deriving high-order A-stable RKN methods based on existing RK schemes.
Abstract: This work intends to contribute to the growing literature on the direct numerical solution of special and general second-order initial value problems (IVPs) of ordinary differential equations (ODEs) by reformulating an A-stable Runge-Kutta (RK) method as a type of implicit Runge–Kutta–Nystrӧm methods (RKN). Unlike traditional methods that reduce se...
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