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We study an electrodynamics consistent with anisotropic transformations of space-time with an arbitrary dynamic exponent z. The equations of motion and conserved quantities are explicitly obtained. We show that the propagator of this theory can be regarded as a quantum correction to the usual propagator. Moreover we obtain that both the momentum and angular momentum are not modified, but their conservation laws do change. We also show that in this theory the speed of light and the electric charge are modified with z. The magnetic monopole in this electrodynamics and its duality transformations are also investigated. For that we found that there exists a dual electrodynamics, with higher derivatives in the electric field, invariant under the same anisotropic transformations.

Recently the liquidity of financial markets and transaction costs have become a topic of great interest in financial risk management. In this paper, a hypotetical nonlinear model of option pricing that occurs when the effects of market illiquidity and transaction costs are taken into account and an approximate solution is obtained through the Adomian decomposition method. Finally, two numerical examples are investigated to demonstrate the efficiency of our approach.

This paper retrieves an optical 1–soliton solution to a model that is written as a concatenation of the Lakshmanan–Porsezian–Daniel model and Sasa–Satsuma equation. The method of undetermined coefficients obtains a full spectrum of 1–soliton solutions. The multiplier approach yields the conserved densities, which subsequently lead to the conserved quantities from the bright 1–soliton solution.

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In the present work we consider the mathematical model that describes brain tumour growth (glioblastomas) under medical treatment. Based on the medical study presented by R. Stupp et al. (New Engl Journal of Med 352: 987-996, 2005) which evidence that, combined therapies such as, radiotherapy and chemotherapy, produces negative tumour-growth, and using the mathematical model of P. K. Burgess et al. (J Neuropath and Exp Neur 56: 704-713, 1997) as an starting point, we present a model for tumour growth under medical treatment represented by a non-linear partial differential equation that is solved using the Adomian Decomposition Method (ADM). It is also shown that the non-linear term efficiently models the effects of the combined therapies. By means of a proper use of parameters, this model could be used for calculating doses in radiotherapy and chemotherapy.

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