Hyperbolic Heat Transfer Equation Research Articles

Chaotic finite difference operators.

In this article, we analyze the chaotic behavior of finite difference operators associated with certain differential equations. Our examples range from numerical schemes for a birth-and-death model with proliferation to a class of second-order partial differential equations that includes the hyperbolic heat transfer equation, the telegraph equation, and the wave equation. We provide sufficient conditions on the spatial and time steps of the scheme that guarantee chaos for the corresponding operators, and we compare them with the conditions needed to ensure chaotic analytical solutions.

Analytical solution of heat-transfer in central part of tibia bone tissue using non-Fourier heat equation with a laser heat source

Hyperbolic heat transfer equation considering a laser heat source is analytically investigated using Laplace transform method to study the temperature distribution in the central part of the tibia bone tissue. The target zone is the compact layer of tibia, and the laser irradiation in four near-surface points during 2 s is taken into account. Holmium–Yttrium–Aluminum-Garnet (Ho-YAG) laser is utilized in the simulations. Results indicate that there is a trilateral relationship among the temperature changes of various points in the tissue, the power used and the wavelength. While neglecting the effects of blood circulation upon laser interaction with the bone surface, the results reveal that the temperature of the points adjacent to the bone surface dramatically increases up to 85 °C. Meanwhile, the heat source is provided based on the Beer-Lambert law, and finally it demonstrates that the penetration depth exponentially reduces the intensity and causing considerable decrease in temperature. For the first time, the non-Fourier heat transfer equation was solved using the Laplace transform method for the tibia bone, and its results were compared with the numerical solution results using the Leap- Frog method. Knowing the temperature changes caused by laser inside the tissue is very crucial in Laser tissue welding, tissue ablation, photocoagulation and photo-thermal denaturation, angioplasty, treatment of granulation tissues, change and deformation of cartilage.

On Certain Applications of the Hyperbolic Heat Transfer Equation and Methods for Its Solution

When creating new technological processes based on the use of high-intensity energy flows, it is necessary to take into account the finite speed of heat transfer. This can be done using the hyperbolic heat transfer equation obtained by A. V. Lykov within the framework of nonequilibrium phenomenological thermodynamics as a consequence of a generalization of the Fourier law for flows and the equation of thermal balance. In previous works, V. N. Khankhasaev modeled the process of switching off an electric arc in a gas flow using this equation. In this paper, we present a mathematical model of this process including the period of stable burning of an electric arc until the shutdown moment, which consists of the replacement of the strongly hyperbolic heat transfer equation by a hyperbolicparabolic equation. For the mixed heat transfer equation obtained, we state certain boundary-value problems, solve them by numerical algorithms, and obtain temperature fields that are consistent with the available experimental data.

The time fractional approach for the modeling of thermal therapies: Temperature analysis in laser irradiation

In this study we assessed the use of the fractional derivative formulation of the heat transmission equation (FDHTE) as an alternative to the classical or parabolic heat transfer equation (PHTE) in the mathematical modeling of some thermal therapy processes. We obtained the FDHTE analytical solutions in two cases: a general case of heat transfer in a finite bar with different and constant temperature in its extremes (without heat source), and the heating by a laser source of a semi-infinite medium which includes a heat source and it was considered in thermal therapies to destroy or alter biological tissue. Both solutions were obtained analytically and compared with the PHTE results. We also compared the FDHTE solution with the results of the hyperbolic heat transfer equation (HHTE), which is another alternative to the PHTE, but is only used for problems in which intense heat is applied to materials for very short times. The results show that the FDHTE can be used as an alternative to the PHTE in thermal therapy processes in which the PHTE theoretical models underestimate or overestimate the temperatures achieved in tissue. Unlike the HHTE, the FDHTE is not restricted to special problems only. We have thus laid the groundwork for the analytical resolution of the problems considered by the FDHTE.

On some finite difference schemes for solution of hyperbolic heat conduction problems

Abstract We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem for hyperbolic heat transfer equation. New stability and approximation error estimates are proved and it is noted that some statements given in the above papers should be modified and improved. Finally, two robust finite difference schemes are proposed, that can be used for both, the hyperbolic and parabolic heat transfer equations. Results of numerical experiments are presented.

CHAOTIC ASYMPTOTIC BEHAVIOR OF THE HYPERBOLIC HEAT TRANSFER EQUATION SOLUTIONS

We describe the periodic solutions and study the chaotic behavior of the solution semigroup of a Cauchy problem corresponding to the hyperbolic heat transfer equation.

Anti-diffusive methods for hyperbolic heat transfer

The hyperbolic heat transfer equation is a model used to replace the Fourier heat conduction for heat transfer of extremely short time duration or at very low temperature. Unlike the Fourier heat conduction, in which heat energy is transferred by diffusion, thermal energy is transferred as wave propagation at a finite speed in the hyperbolic heat transfer model. Therefore methods accurate for Fourier heat conduction may not be suitable for hyperbolic heat transfer. In this paper, we present two anti-diffusive methods, a second-order TVD-based scheme and a fifth-order WENO-based scheme, to solve the hyperbolic heat transfer equation and extend them to two-dimension, including a nonlinear application caused by temperature-dependent thermal conductivity. Several numerical examples are applied to validate the methods. The current solution is compared in one-dimension with the analytical one as well as the one obtained from a high-resolution TVD scheme. Numerical results indicate that the fifth-order anti-diffusive method is more accurate than the high-resolution TVD scheme and the second-order anti-diffusive method in solving the hyperbolic heat transfer equation.

Modeling the heating of biological tissue based on the hyperbolic heat transfer equation

In modern surgery, a multitude of minimally intrusive operational techniques are used which are based on the point heating of target zones of human tissue via laser or radiofrequency currents. Traditionally, these processes are modeled by the bioheat equation introduced by Pennes, who considers Fourier’s theory of heat conduction. We present an alternative and more realistic model established using the hyperbolic equation of heat transfer. To demonstrate some features and advantages of our proposed method, we apply the results obtained to different types of tissue heating with high energy fluxes, in particular radiofrequency heating and pulsed laser treatment of the cornea to correct refractive errors. We hope that the results from our approach will help with refining surgical interventions in this novel field of medical treatment.

Thermal modeling for pulsed radiofrequency ablation: Analytical study based on hyperbolic heat conduction

The objectives of this study were to model the temperature progress of a pulsed radiofrequency (RF) power during RF heating of biological tissue, and to employ the hyperbolic heat transfer equation (HHTE), which takes the thermal wave behavior into account, and compare the results to those obtained using the heat transfer equation based on Fourier theory (FHTE). A theoretical model was built based on an active spherical electrode completely embedded in the biological tissue, after which HHTE and FHTE were analytically solved. We found three typical waveforms for the temperature progress depending on the relations between the dimensionless duration of the RF pulse delta(a) and the expression square root of lambda(rho-1), with lambda as the dimensionless thermal relaxation time of the tissue and rho as the dimensionless position. In the case of a unique RF pulse, the temperature at any location was the result of the overlapping of two different heat sources delayed for a duration delta(a) (each heat source being produced by a RF pulse of limitless duration). The most remarkable feature in the HHTE analytical solution was the presence of temperature peaks traveling through the medium at a finite speed. These peaks not only occurred during the RF power switch-on period but also during switch off. Finally, a physical explanation for these temperature peaks is proposed based on the interaction of forward and reverse thermal waves. All-purpose analytical solutions for FHTE and HHTE were obtained during pulsed RF heating of biological tissues, which could be used for any value of pulsing frequency and duty cycle.

Analytical thermal-optic model for laser heating of biological tissue using the hyperbolic heat transfer equation

In this paper, we solve in an analytical way the thermal-optic coupled problem associated with a 1D model of non-perfused homogeneous biological tissue irradiated by a laser beam. We consider a laser pulse duration of 200 micros and study the temperatures of areas very close to the point of laser beam application. We consider that these values of the temporal and spatial variables mean that the problem has to be solved by means of the hyperbolic heat conduction model instead of the classic or parabolic model. We therefore obtain the solution using both models and apply the temperature profiles obtained to a specific biological tissue for comparison. Finally, we theoretically study the effect of the thermal relaxation time on the temperature profiles in the tissue for both heating and cooling phases (i.e. during and after laser application).

Assessment of Hyperbolic Heat Transfer Equation in Theoretical Modeling for Radiofrequency Heating Techniques

Theoretical modeling is a technique widely used to study the electrical-thermal performance of different surgical procedures based on tissue heating by use of radiofrequency (RF) currents. Most models employ a parabolic heat transfer equation (PHTE) based on Fourier’s theory, which assumes an infinite propagation speed of thermal energy. We recently proposed a one-dimensional model in which the electrical-thermal coupled problem was analytically solved by using a hyperbolic heat transfer equation (HHTE), i.e. by considering a non zero thermal relaxation time. In this study, we particularized this solution to three typical examples of RF heating of biological tissues: heating of the cornea for refractive surgery, cardiac ablation for eliminating arrhythmias, and hepatic ablation for destroying tumors. A comparison was made of the PHTE and HHTE solutions. The differences between their temperature profiles were found to be higher for lower times and shorter distances from the electrode surface. Our results therefore suggest that HHTE should be considered for RF heating of the cornea (which requires very small electrodes and a heating time of 0.6 s), and for rapid ablations in cardiac tissue (less than 30 s).

Effect of the thermal wave in radiofrequency ablation modeling: an analytical study

To date, all radiofrequency heating (RFH) theoretical models have employed Fourier's heat transfer equation (FHTE), which assumes infinite thermal energy propagation speed. Although this equation is probably suitable for modeling most RFH techniques, it may not be so for surgical procedures in which very short heating times are employed. In such cases, a non-Fourier model should be considered by using the hyperbolic heat transfer equation (HHTE). Our aim was to compare the temperature profiles obtained from the FHTE and HHTE for RFH modeling. We built a one-dimensional theoretical model based on a spherical electrode totally embedded and in close contact with biological tissue of infinite dimensions. We solved the electrical–thermal coupled problem analytically by including the power source in both equations. A comparison of the analytical solutions from the HHTE and FHTE showed that (1) for short times and locations close to the electrode surface, the HHTE produced temperatures higher than the FHTE, however, this trend became negligible for longer times, when both equations produced similar temperature profiles (HHTE always being higher than FHTE); (2) for points distant from the electrode surface and for very short times, the HHTE temperature was lower than the FHTE, however, after a delay time, this tendency inverted and the HHTE temperature increased to the maximum; (3) from a mathematical point of view, the HHTE solution showed cuspidal-type singularities, which were materialized as a temperature peak traveling through the medium at a finite speed. This peak rose at the electrode surface, and clearly reflected the wave nature of the thermal problem; (4) the differences between the FHTE and HHTE temperature profiles were smaller for the lower values of thermal relaxation time and locations further from the electrode surface.

THE RELATIONSHIP BETWEEN THE LOCAL TEMPERATURE AND THE LOCAL HEAT FLUX WITHIN A ONE-DIMENSIONAL SEMI-INFINITE DOMAIN FOR THE DUAL-LAG MODEL OF HEAT CONDUCTION

The relationship between the local temperature and the local heat flux has been established for the homogeneous heat equation, which takes into account the dual-phase-lag. This relationship has been written in the form of a convolution integral, involving the modified Bessel functions. Thus, the integral equation obtained has been solved numerically for processes of surface heatingwith time scale in the order of picoseconds. The results have been compared with the results obtained from the classical equation of heat conduction (parabolic heat transfer), the hyperbolic heat transfer equation with single-phase-lag, and the numerical solution of the heat equation with dual-phase-lag, which were obtained using the finite difference method.

Theory of normal-zone propagation along a superconducting wire with allowance for thermal-relaxation processes

For high velocity of a longitudinal propagation of a normal zone along a superconducting wire, one should take into account heat-flux relaxation processes. In such cases these waves are described by a hyperbolic heat-transfer equation, which involves not only the heat-source function but also its derivative. The relaxation effects lead to a limitation of possible values of the normal-zone velocities and to the nonuniqueness of wave solutions. Moreover, the relationships that determine the temperature field in a wire and the propagation speed of the normal zone are obtained. The transient boundary condition between the wire and substrate (e.g., helium bath) is deduced from a heat-flux relaxation theory.

Molecular-kinetics basis for the heat-transfer equation

A hyperbolic heat-transfer equation is derived by the Maxwell method. The correlation between the components of the thermal velocities of the atoms or molecules is taken into account.