This article presents a preliminary study of the longitudinal self-compression of ultra-intense Gaussian laser pulse in a magnetized plasma, when relativistic nonlinearity is active. This study has been carried out in 1D geometry under a nonlinear Schrodinger equation and higher-order paraxial (nonparaxial) approximation. The nonlinear differential equations for self-compression and self-focusing have been derived and solved by the analytical and numerical methods. The dielectric function and the eikonal have been expanded up to the fourth power of r (radial distance). The effect of initial parameters, namely incident laser intensity, magnetic field, and initial pulse duration on the compression of a relativistic Gaussian laser pulse have been explored. The results are compared with paraxial-ray approximation. It is found that the compression of pulse and pulse intensity of the compressed pulse is significantly enhanced in the nonparaxial region. It is observed that the compression of the high-intensity laser pulse depends on the intensity of laser beam (a0), magnetic field (ω c ), and initial pulse width (τ0). The preliminary results show that the pulse is more compressed by increasing the values of a0, ω c , and τ0.

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Boundary elements have emerged as a powerful alternative to finite elements particularly in cases where better accuracy is required. The most important features of boundary elements however is that it only requires descretization of the surface rather than the volume. Here, A general algorithm of the boundary integral method has been formulated for solving elliptic partial differential equations. The broad applicability of the approach is illustrated with a problem of practical interest giving the solution of the Laplace equation for potential flow with mixed boundary problems. The results and patterns are shown in tables and figures and compared well with Brebbia [1] are found in good agreement.