Articles written in Pramana – Journal of Physics

    • Periodic solution of the cubic nonlinear Klein–Gordon equation and the stability criteria via the He-multiple-scales method


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      The current work demonstrated a new technique to improve the accuracy and computational efficiency of the nonlinear partial differential equation based on the homotopy perturbation method (HPM). In this proposal, two different homotopy perturbation expansions, the outer expansion and the inner one, are introduced based on two different homotopy parameters. The multiple-scale homotopy technique (He-multiple-scalas method) is applied as an outer perturbation for the nonlinear Klein–Gordon equation.A highly accurate periodic temporal solution has been derived from three orders of perturbation. The amplitude equation, which is imposed as a uniform condition, is of the fourth-order cubic–quintic nonlinear Schrödinger equation. The standard HPM with another homotopy parameter has been used as an inner perturbation to obtain a spatial solution of the nonlinear Schrödinger equation. The cubic– quintic Landau equation is obtained in the inner perturbation technique. Finally, the approximate solution is derived from the temporal and spatial solutions. Further, two different tools are used to obtain the same stability conditions. One of them is a new tool based on the HPM, by constructing the nonlinear frequency. The method adopted here is important and powerful for solving partial differential nonlinear oscillator systems arising in nonlinear science and engineering.

    • A novelty to the nonlinear rotating Rayleigh–Taylor instability


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      This paper presents a novel approach for studying the nonlinear Rayleigh–Taylor instability (RTI). The system deals with two rotating superposed infinite hydromagnetic Darcian flows through porous media under the influence of a uniform tangential magnetic field. The field allows the presence of currents on the surface of separation. The appropriate linear governing equations are solved and confirmed with the corresponding nonlinear boundary conditions. A nonlinear characteristic of the surface deflection is deducted. Away from the traditional techniques of the stability analysis, the work introduces a new one. The analysis depends mainly on the homotopy perturbation method (HPM). To achieve an analytical approximate periodic solution of the surface deflection, the secular terms are removed. This cancellation resulted in well-known amplitude equations. These equations are utilised to achieve stability criteria of the system. Therefore, the stability configuration is exercised in linear aswell as nonlinear approaches. The mathematical procedure adopted here is simple, promising and powerful. The method may be used to treat more complicated nonlinear differential equations that arise in science, physics andengineering applications. A numerical calculation is performed to graph the implication of various parameters on the stability picture. In addition, for more convenience, the surface deflection is depicted.

    • Modified multiple scale technique for the stability of the fractional delayed nonlinear oscillator


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      In the present proposal, the familiar method of the parameter expansion is combined with the multiple scales to study the stability behaviour of the Riemann–Liouville fractional derivative applied to the cubic delayed Duffing oscillator. The analysis of the modified multiple scale perturbation leads to a system of nonlinear differential-algebraic equations governing the solvability conditions. The nonlinear differential equation was reduced to the linear differential equation with the help of the algebraic one. The stability attitude of the periodic motion is determined by the steady-state analysis. Such a periodic motion is needed to better understand the dynamics of the fractional cubic delayed Duffing oscillator.

    • Nonlinear dynamical analysis of a time-fractional Klein–Gordon equation


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      In the present work, an enhanced perturbation analysis to solve a time-fractional Klein–Gordon equation (KG equation) and obtain an analytic approximate periodic solution is examined. The Riemann–Liouville fractionalderivative is utilised. A travelling wave solution is adopted throughout the perturbation method by including two small perturbation parameters. The amplitude equation is formulated in the form of a cubic–quintic complexnonlinear Schrödinger equation. The solution of this equation leads to a transcendental frequency equation. An approximate solution to this frequency equation is performed. The stability criteria are derived. The procedure adopted here is very significant and powerful for solving many nonlinear partial differential equations (NLPDEs) arising in nonlinear science and engineering.

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