This paper presents the spectral and light curve analysis of gamma-ray burst (GRB) afterglows, focusing on synchrotron emission from a relativistic shock. The afterglow is modeled as synchrotron radiation from a power-law distribution of electrons in an expanding relativistic shell. The spectrum and light curve consist of several power-law segments. The light curve is calculated under two hydrodynamical models: fully adiabatic and fully radiative. The results are compared with observations of GRB afterglows. The synchrotron spectrum is derived from the emission of relativistic electrons in a magnetic field. The spectral power varies as ν^{1/3} for ν < ν(γ_e) and cuts off exponentially for ν > ν(γ_e). The peak power occurs at ν(γ_e), where it is approximately proportional to γB. The critical electron Lorentz factor γ_c determines whether the electrons cool significantly due to synchrotron radiation. The spectrum is divided into three segments: a low-energy tail, a power-law segment, and an exponential cutoff. The paper considers two cases: fast cooling (γ_m > γ_c) and slow cooling (γ_c > γ_m). In the fast cooling case, all electrons cool to γ_c, while in the slow cooling case, only electrons with γ_e > γ_c cool. The flux at a given frequency is derived based on these cases. The hydrodynamical evolution of the shock is considered in two extreme limits: adiabatic and radiative. In the adiabatic case, the shock energy is constant, while in the radiative case, the energy varies with time. The light curve is calculated based on the temporal evolution of the break frequencies ν_c and ν_m, and the peak flux F_ν,max. The results show that the peak flux is constant in the slow cooling limit for adiabatic evolution. In the fast cooling limit, the peak flux decreases with time as F_ν,max ∝ t^{-3/7}, and the peak frequency varies as ν_c ∝ t^{-2/7}. The light curve has different slopes depending on the evolution regime. The paper also discusses the spectral index β, which is related to the light curve slope α. The results are consistent with observations of GRB afterglows, showing different values of α in the optical and X-ray bands. The paper concludes that the observed decay of GRB afterglows with α ≈ 1.2 or 1.4 supports the shock model and adiabatic assumption. The analysis provides a framework for understanding the spectral and light curve evolution of GRB afterglows.This paper presents the spectral and light curve analysis of gamma-ray burst (GRB) afterglows, focusing on synchrotron emission from a relativistic shock. The afterglow is modeled as synchrotron radiation from a power-law distribution of electrons in an expanding relativistic shell. The spectrum and light curve consist of several power-law segments. The light curve is calculated under two hydrodynamical models: fully adiabatic and fully radiative. The results are compared with observations of GRB afterglows. The synchrotron spectrum is derived from the emission of relativistic electrons in a magnetic field. The spectral power varies as ν^{1/3} for ν < ν(γ_e) and cuts off exponentially for ν > ν(γ_e). The peak power occurs at ν(γ_e), where it is approximately proportional to γB. The critical electron Lorentz factor γ_c determines whether the electrons cool significantly due to synchrotron radiation. The spectrum is divided into three segments: a low-energy tail, a power-law segment, and an exponential cutoff. The paper considers two cases: fast cooling (γ_m > γ_c) and slow cooling (γ_c > γ_m). In the fast cooling case, all electrons cool to γ_c, while in the slow cooling case, only electrons with γ_e > γ_c cool. The flux at a given frequency is derived based on these cases. The hydrodynamical evolution of the shock is considered in two extreme limits: adiabatic and radiative. In the adiabatic case, the shock energy is constant, while in the radiative case, the energy varies with time. The light curve is calculated based on the temporal evolution of the break frequencies ν_c and ν_m, and the peak flux F_ν,max. The results show that the peak flux is constant in the slow cooling limit for adiabatic evolution. In the fast cooling limit, the peak flux decreases with time as F_ν,max ∝ t^{-3/7}, and the peak frequency varies as ν_c ∝ t^{-2/7}. The light curve has different slopes depending on the evolution regime. The paper also discusses the spectral index β, which is related to the light curve slope α. The results are consistent with observations of GRB afterglows, showing different values of α in the optical and X-ray bands. The paper concludes that the observed decay of GRB afterglows with α ≈ 1.2 or 1.4 supports the shock model and adiabatic assumption. The analysis provides a framework for understanding the spectral and light curve evolution of GRB afterglows.