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a b s t r a c tIn this paper, He's Frequency-Amplitude Formulation (HFAF) and He's Energy Balance Method (HEBM) are employed to solve the generalized Duffing equation in the form ofFor any arbitrary power of n, a frequency analysis is carried out and the relationship between the natural frequency and the initial amplitude is obtained in analytical form. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HFAF, HEBM and exact integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations are achieved between the numerical results.

Hyperlipidemia can lead to atherosclerosis by lipoprotein deposition inside the vessel wall and oxidative stress induction that leads to the formation of atherosclerotic plaque. Oxidized low-density lipoprotein particles (Ox-LDL) have a key role in the pathogenesis of atherosclerosis. The lipid-lowering properties and antioxidants of the grape seed can be beneficial in atherosclerosis prevention. We conducted a randomized double-blind placebo-controlled crossover clinical trial. Fifty-two mildly hyperlipidemic individuals were divided into two groups that received either 200 mg/day of the red grape seed extract (RGSE) or placebo for 8 weeks. After an 8-week washout period, the groups were crossed over for another 8 weeks. Lipid profiles and Ox-LDL were measured at the beginning and the end of each phase. RGSE consumption reduced total cholesterol (-10.68±26.76 mg/dL, P=.015), LDL cholesterol (-9.66±23.92 mg/dL, P=.014), and Ox-LDL (-5.47±12.12 mg/dL, P=.008). While triglyceride and very low-density lipoprotein cholesterol were decreased and high-density lipoprotein cholesterol was increased by RGSE, the changes were not statistically significant. RGSE consumption decreases Ox-LDL and has beneficial effects on lipid profile-consequently decreasing the risk of atherosclerosis and cardiovascular disorders-in mild hyperlipidemic individuals.