This study addresses a class of inhomogeneous delay differential equations characterized by variable coefficients and the presence of two proportional delays. An exponential-type transformation is employed to convert the inhomogeneous formulation into an equivalent homogeneous pantograph equation. This reformulation enables the derivation of an explicit analytical solution represented by a superposition of an exponential function and a power series with coefficients given in closed product form. A detailed investigation of the convergence properties of the associated infinite product and series is conducted, yielding clear criteria for global convergence. Furthermore, specific parameter configurations are identified under which the series solution reduces to a finite sum, leading to exact analytical expressions. As an important application, the inhomogeneous Ambartsumian equation involving two proportional delays is examined, and new closed-form and exact solutions are established. Numerical experiments are included to support the theoretical analysis and to demonstrate the rapid convergence and the practical effectiveness of the proposed solution approach.
Citation: Essam R. El-Zahar, Abdelhalim Ebaid, Laila F. Seddek, A. M. Altamimi. Exact and series solutions for a class of inhomogeneous delay equations with variable coefficients and two proportional delays[J]. AIMS Mathematics, 2026, 11(6): 19162-19176. doi: 10.3934/math.2026780
This study addresses a class of inhomogeneous delay differential equations characterized by variable coefficients and the presence of two proportional delays. An exponential-type transformation is employed to convert the inhomogeneous formulation into an equivalent homogeneous pantograph equation. This reformulation enables the derivation of an explicit analytical solution represented by a superposition of an exponential function and a power series with coefficients given in closed product form. A detailed investigation of the convergence properties of the associated infinite product and series is conducted, yielding clear criteria for global convergence. Furthermore, specific parameter configurations are identified under which the series solution reduces to a finite sum, leading to exact analytical expressions. As an important application, the inhomogeneous Ambartsumian equation involving two proportional delays is examined, and new closed-form and exact solutions are established. Numerical experiments are included to support the theoretical analysis and to demonstrate the rapid convergence and the practical effectiveness of the proposed solution approach.
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Essam R. El-Zahar, Abdelhalim Ebaid, Laila F. Seddek, A. M. Altamimi. Exact and series solutions for a class of inhomogeneous delay equations with variable coefficients and two proportional delays[J]. AIMS Mathematics, 2026, 11(6): 19162-19176. doi: 10.3934/math.2026780
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