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In the present study, an axisymmetric one-dimensional Pennes bioheat transfer model is used to investigate analytically the heat transport in the peripheral region of a human limb undergoing healing after surgery. The analytical exact solution to the model is presented in terms of modified Bessel functions. Through the found analytical solution, the time-dependent temperature distribution of normal and abnormal tissues is investigated. Effects of different biological parameters like metabolic heat generation, rate of sweat evaporation, variable physiological parameters in dermal layer of peripheral region, and ambient temperature on temperature distribution are also studied.

Motivated by recent experimental investigations in the context of matter wave solitons in Bose-Einstein condensates (BECs), we consider the 1+1 Gross-Pitaevskii equation with complex time-varying harmonic potential, and time-varying cubic and quintic nonlinearities. By performing a modified lens-type transformation for the one-dimensional GP equation, we present one and/or two parameter exact analytical solutions which describe the propagation of bright, kink, and dark solitary waves on the vanishing continuous wave (cw) background. Based on exact analytical solutions of the GP equation, we investigate analytically the dynamics of matter-wave solitons in the BEC systems. Our studies show that the solitons’ amplitude depends on both the scattering length and the feeding/loss term of the potential while their motion depends on the external trapping potential and solution parameters.

We consider the one-dimensional bio-heat transfer problems with linear temperature-dependent blood perfusion and spatially distributed heating which describe heat transport in blood perfused tissues. Analytical methods for solving nonlinear partial differential equations are combined with the Crank-Nicholson scheme to study several selected typical bio-heat transfer processes, which are often encountered in cancer hyperthermia, laser surgery, and thermal comfort analysis. The results mainly show that: i) the larger heating power increases the amplitude of the temperature response of the tissues; ii) a very low frequency of the heating power can be associated to a large frequency of the surrounding medium temperature to make irregular the frequency of the resulted temperature response.

Analytical entire traveling wave solutions to the 1+1 density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method are presented in this paper. This equation can be regarded as an extension case of the Fisher-Kolmogoroff equation, which is used for studying insect and animal dispersal with growth dynamics. The analytical solutions are then used to investigate the effect of equation parameters on the population distribution.

We introduce a phase imprint into the order parameter describing the influence of blood flow on the temperature distribution in the tissue described by the one-dimensional Pennes equation and then engineer the imprinted phase suitably to generate a modified Pennes equation with a gradient term (known in the theory of biological systems as convective term) which is associated with the heat convected by the flowing blood. Using the derived model, we analytically investigate temperature distribution in biological tissues subject to two different spatial heating methods. The applicability of our results is illustrated by one of typical bio-heat transfer problems which is often encountered in therapeutic treatment, cancer hyperthermia, laser surgery, thermal injury evaluation, etc. Analyzing the effect of the convective term on temperature distribution, we found that an optimum heating of a biological system can be obtained through regulating the convective term.

A quintic nonlinear Schrodinger (NLS) equation, with derivative cubic terms, that governs the propagation of nonlinear signals in a nonlinear transmission line (NLTL) is considered. By combining a special phase-imprint transformation with a modified lens transformation, we reduce the equation under consideration to a standard cubic NLS equation with a time-varying gain/loss term and obtain the integrability condition. Under this condition, we first apply a superposition procedure to derive new nonlinear wave signals that propagate with periodic amplitude in the NLTL. Secondly, in the absence of any gain/loss term in the cubic NLS equation, we apply the Darboux transformation to the derived new bright soliton-like signal of the NLTL. For a special form of the gain/loss term of the cubic NLS equation, we combine the homogeneous balance principle and an F-expansion technique to show the propagation of both bright and dark soliton-like signals in the NLTL under consideration, and show how to manage the soliton motion in the line.

Several investigators have argued that Pennes’ interpretation of the vascular contribution to heat transfer in perfused tissues fails to account for the actual thermal equilibration process between the flowing blood and the surrounding tissue and proposed new models, presumably based on a more realistic anatomy of the perfused tissue. In this same direction, we introduce an exponential imprint into the thermal order parameter governing the bioheat transfer described by the Pennes model equation and then engineer the imprinted exponent suitably to generate the modified Pennes bioheat transfer model with time-variable coefficients that takes into account the dependence of blood perfusion on both time and temperature. This mathematical model uses blood perfusion to monitor and control temperature distribution in living biological tissues. We observe that all monitoring parameters interact to contribute to the control and monitoring of temperature distribution in tissues. Employing similarity and F-expansion approaches, we propose an algorithm for finding thermal traveling wave-like solutions of the model.