Abstract
The general obstacle framework has found applications in steady state fluid interaction, thin-plate fluid dynamics, study of minimal surfaces, control theory, elasto-statics, etc. The obstacle problem involving the fractional operator indeed appears in many contexts, such as in the analysis of anomalous diffusion, in the quasi-geostrophic flow problem, and in pricing of American options regulated by assets evolving in relation to jump processes; these notable applications in financial mathematics and physics made the obstacle problem very important in recent times. In this work, we present a fractional contact problem in which derivative of fractional order in the sense of Caputo is involved. Using the penalty function method, we degenerate it into a system of fractional boundary value problems with known obstacle. We apply the variational iteration method (VIM) for finding the series solution of these fractional BVPs. In order to ensure the accuracy and convergence of solution, residual errors of the solutions for various values of fractional parameters are plotted. The quite accurate results show that variational iteration method is one of the highly potential and robust method for solving fractional BVPs.
Highlights
The obstacle problem is significant in the field of variational inequalities
The main concern is to evaluate the equilibrium position of elastic membrane lying over a given obstacle
Several problems in engineering sciences can be modelled as obstacle problems, see Refs. 14, 34, and 39
Summary
The obstacle problem is significant in the field of variational inequalities. In obstacle problems, the main concern is to evaluate the equilibrium position of elastic membrane lying over a given obstacle. Notable efforts have been made to establish strong and stable numerical and analytical techniques for solving fractional differential equations of physical interest. Kumar et al. presented a hybrid numerical scheme based on the homotopy analysis transform method (HATM) to examine the fractional model of nonlinear wave-like equations having variable coefficients, Goswami used homotopy perturbation sumudu transform method (HPSTM) to solve fractional equal width, modified equal width equations, Singh studied the q-fractional homotopy analysis transform method (q-FHATM). To find the analytical and approximate solutions of space-time arbitrary order advection-dispersion equations with nonlocal effects, Martin proposed an algorithm based on variational iteration method and Laplace transforms for solving a fractional differential equations and presented the stability of fractional operator defined by VIM. Many researchers discussed fractional type contact problems based on obstacle due to its wide applications in physics and engineering
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