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A modified He’s homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. The He’s homotopy perturbation method is modified by truncating the infinite series corresponding to the first order approximate solution before introducing this solution in the second order linear differential equation, and so on. We find this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. The approximate formulas obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient.

A modified He's homotopy perturbation method is used to calculate the periodic solutions of a nonlinear pendulum. The method has been modified by truncating the infinite series corresponding to the first-order approximate solution and substituting a finite number of terms in the second-order linear differential equation. As can be seen, the modified homotopy perturbation method works very well for high values of the initial amplitude. Excellent agreement of the analytical approximate period with the exact period has been demonstrated not only for small but also for large amplitudes A (the relative error is less than 1% for A < 152°). Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.

A modified He’s homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. The He’s homotopy perturbation method is modified by truncating the infinite series corresponding to the first order approximate solution before introducing this solution in the second order linear differential equation, and so on. We find this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. The approximate formulas obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient.

A modified He's homotopy perturbation method is used to calculate the periodic solutions of a nonlinear pendulum. The method has been modified by truncating the infinite series corresponding to the first-order approximate solution and substituting a finite number of terms in the second-order linear differential equation. As can be seen, the modified homotopy perturbation method works very well for high values of the initial amplitude. Excellent agreement of the analytical approximate period with the exact period has been demonstrated not only for small but also for large amplitudes A (the relative error is less than 1% for A < 152°). Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.

The coupled wave theory of Kogelnik has given a well-established basis for the comprehension of how light propagates inside a volume hologram. This theory gives a good approximation for the diffraction efficiency of both volume phase holograms and volume absorption holograms. Mixed holograms (phase and absorption) have also been dealt with from the point of view of the coupled wave theory. In this paper we use Kogelnik's coupled wave theory to give quantitative information about the mechanisms which produce mixed gratings in photographic emulsions. In particular, we demonstrate that mixed amplitude-phase gratings are recorded on photographic emulsions when fixation-free bleaching techniques are used to obtain volume holograms. We will prove that the oxidation products of the bleach can give rise to an absorption modulation at high values of exposure and high concentrations of potassium bromide in the bleach bath. We will also give quantitative data regarding the absorption created by these oxidation products.

The coupled wave theory of Kogelnik has given a well-established basis for the comprehension of how light propagates inside a volume hologram. This theory gives a good approximation for the diffraction efficiency of both volume phase holograms and volume absorption holograms. Mixed holograms (phase and absorption) have also been dealt with from the point of view of the coupled wave theory. In this paper we use Kogelnik's coupled wave theory to give quantitative information about the mechanisms which produce mixed gratings in photographic emulsions. In particular, we demonstrate that mixed amplitude-phase gratings are recorded on photographic emulsions when fixation-free bleaching techniques are used to obtain volume holograms. We will prove that the oxidation products of the bleach can give rise to an absorption modulation at high values of exposure and high concentrations of potassium bromide in the bleach bath. We will also give quantitative data regarding the absorption created by these oxidation products.