Exact Solution for a Class of First-Order Reflected Equations Involving a Polynomial-Type Forcing Term of Arbitrary Finite Degree
- 1Department of Mathematics, College of Science, Majmaah University, Al-Majmaah 11952, Saudi Arabia
Abstract
Differential equations involving reflection of the argument arise in numerous mathematical models where the present state is coupled with its symmetric counterpart. In this paper, we investigate a challenging reflected equation, expressed as y′(t)=ay(t)+by(−t)+∑k=0mcktk, where the forcing function is an arbitrary polynomial of finite degree. A constructive transformation is developed that converts the nonhomogeneous reflected equation into a homogeneous reflection equation. Explicit closed-form expressions for the transformation coefficients are derived and proved by backward induction. Consequently, the original problem is reduced to a homogeneous reflected model whose exact solution is obtained in power-series form and subsequently expressed through hyperbolic or trigonometric functions according to the sign of a2−b2. To complete the analysis, a reduction-of-order technique is employed to establish an equivalent second-order ordinary differential equation without reflection. This approach enables the derivation of exact solutions in the singular cases a=b and a=−b, where the transformation method is no longer applicable. The obtained results provide a unified analytical methodology for a broad class of first-order reflection equations with polynomial forcing and furnish exact closed-form solutions for both regular and resonant parameter regimes.
Related
- Conditional Reflected Backward Stochastic Differential Equations
- Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations
- Constrained Mean Variance Portfolio Selection under Reflected Factor Dynamics: Stochastic Riccati Equations, Neumann Problems,and a Deep Solver
- A Polynomial-Time Attack on the McEliece Cryptosystem on Elliptic Codes with Arbitrary Divisors
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
BibTeX
@article{albidah2026exact,
title = {{Exact Solution for a Class of First-Order Reflected Equations Involving a Polynomial-Type Forcing Term of Arbitrary Finite Degree}},
author = {Abdulrahman B. Albidah},
journal = {Axioms},
year = {2026},
month = sep,
doi = {10.3390/axioms15100708},
url = {https://doi.org/10.3390/axioms15100708},
note = {CC BY},
}
RIS
TY - JOUR
TI - Exact Solution for a Class of First-Order Reflected Equations Involving a Polynomial-Type Forcing Term of Arbitrary Finite Degree
AU - Abdulrahman B. Albidah
PY - 2026
DA - 2026/09/01
JO - Axioms
DO - 10.3390/axioms15100708
UR - https://doi.org/10.3390/axioms15100708
AB - Differential equations involving reflection of the argument arise in numerous mathematical models where the present state is coupled with its symmetric counterpart. In this paper, we investigate a challenging reflected equation, expressed as y′(t)=ay(t)+by(−t)+∑k=0mcktk, where the forcing function is an arbitrary polynomial of finite degree. A constructive transformation is developed that converts the nonhomogeneous reflected equation into a homogeneous reflection equation. Explicit closed-form expressions for the transformation coefficients are derived and proved by backward induction. Consequently, the original problem is reduced to a homogeneous reflected model whose exact solution is obtained in power-series form and subsequently expressed through hyperbolic or trigonometric functions according to the sign of a2−b2. To complete the analysis, a reduction-of-order technique is employed to establish an equivalent second-order ordinary differential equation without reflection. This approach enables the derivation of exact solutions in the singular cases a=b and a=−b, where the transformation method is no longer applicable. The obtained results provide a unified analytical methodology for a broad class of first-order reflection equations with polynomial forcing and furnish exact closed-form solutions for both regular and resonant parameter regimes.
SN - 2075-1680
ER -