Journal articles: 'Heat equation'

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N O, Onuoha. "Transformation of Parabolic Partial Differential Equations into Heat Equation Using Hopf Cole Transform." International Journal of Science and Research (IJSR) 12, no. 6 (June 5, 2023): 1741–43. http://dx.doi.org/10.21275/sr23612082710.

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Zeng, Chulan. "Time analyticity of the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations." Communications on Pure & Applied Analysis 21, no. 3 (2022): 749. http://dx.doi.org/10.3934/cpaa.2021197.

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In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>. The potentials include all the nonnegative ones. For the first two equations, we prove if <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> satisfies some growth conditions in <inline-formula><tex-math id="M3">\begin{document}$ (x,t)\in \mathrm{M}\times [0,1] $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> is analytic in time <inline-formula><tex-math id="M5">\begin{document}$ (0,1] $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M6">\begin{document}$ \mathrm{M} $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ R^d $\end{document}</tex-math></inline-formula> or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that <inline-formula><tex-math id="M8">\begin{document}$ u(x,t) $\end{document}</tex-math></inline-formula> is analytic in time at <inline-formula><tex-math id="M9">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general.For the nonlinear heat equation with power nonlinearity of order <inline-formula><tex-math id="M10">\begin{document}$ p $\end{document}</tex-math></inline-formula>, we prove that a solution is analytic in time <inline-formula><tex-math id="M11">\begin{document}$ t\in (0,1] $\end{document}</tex-math></inline-formula> if it is bounded in <inline-formula><tex-math id="M12">\begin{document}$ \mathrm{M}\times[0,1] $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a positive integer. In addition, we investigate the case when <inline-formula><tex-math id="M14">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a rational number with a stronger assumption <inline-formula><tex-math id="M15">\begin{document}$ 0<C_3 \leq |u(x,t)| \leq C_4 $\end{document}</tex-math></inline-formula>. It is also shown that a solution may not be analytic in time if it is allowed to be <inline-formula><tex-math id="M16">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. As a lemma, we obtain an estimate of <inline-formula><tex-math id="M17">\begin{document}$ \partial_t^k \Gamma(x,t;y) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M18">\begin{document}$ \Gamma(x,t;y) $\end{document}</tex-math></inline-formula> is the heat kernel on a manifold, with an explicit estimation of the coefficients.An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable <inline-formula><tex-math id="M19">\begin{document}$ x $\end{document}</tex-math></inline-formula>, implying that the analyticity of space and time can be independent. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.

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Gupta, Rohit, Rakesh Kumar Verma, and Sanjay Kumar Verma. "Solving Wave Equation and Heat Equation by Rohit Transform (RT)." Journal of Physics: Conference Series 2325, no. 1 (August 1, 2022): 012036. http://dx.doi.org/10.1088/1742-6596/2325/1/012036.

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Abstract The wave equation and the heat equation are widely known differential equations coming to light in engineering, basic and material sciences. The differential equations which represent the wave equation and the heat equation are usually solved by the exact technique or by the approximate technique or by the purely numerical technique. Since the implementation of these techniques is very complex, computationally vigorous, and requires elaborate computations, therefore, for finding the solutions of differential equations depicting the wave equation and the heat equation, there is a need to ask for integral transform techniques. Integral transform techniques render productive means for finding the solutions of problems coming to light in engineering, basic and material sciences. The Rohit transform (RT) is a new integral transformation put forward by the author Rohit Gupta recently in the year 2020 and has been utilized for finding the solutions of problems coming to light in engineering, basic and material sciences like other transform techniques. In this study, the RT is brought in for finding the solutions of the heat equation and the wave equation expressed in terms of differential equations which are generally partial in nature.

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Kochubei, Anatoly N., Yuri Kondratiev, and José Luís da Silva. "On fractional heat equation." Fractional Calculus and Applied Analysis 24, no. 1 (January 29, 2021): 73–87. http://dx.doi.org/10.1515/fca-2021-0004.

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Abstract In this paper, the long-time behavior of the Cesaro mean of the fundamental solution for fractional Heat equation corresponding to random time changes in the Brownian motion is studied. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.

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Kiniry, James R., Sumin Kim, and Henri E. Z. Tonnang. "Back to the Future: Revisiting the Application of an Enzyme Kinetic Equation to Maize Development Nearly Four Decades Later." Agronomy 9, no. 9 (September 19, 2019): 566. http://dx.doi.org/10.3390/agronomy9090566.

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With the recent resurgence in interest in models describing maize (Zea mays L.) development rate responses to temperature, this study uses published data to refit the Poikilotherm equation and compare it to broken stick “heat stress” equations. These data were for the development rate of eight open pollinated maize varieties at diverse sites in Africa. The Poikilotherm equation was applied with the original published parameters and after refitting with the data in this study. The heat stress equation was tested after fitting with just the first variety and after fitting with each variety. The Poikilotherm equation with the original parameter values had large errors in predicting development rates in much of the temperature range. The adjusted Poikilotherm equation did much better with the root-mean-square error (RMSE) decreasing from 0.034 to 0.003 (1/day) for a representative variety. The heat stress equation fit to the first variety did better than the Poikilotherm equation when applied to all the varieties. The heat stress equations fitted separately for each variety did not have an improved fit compared to the one heat stress equation. Thus, separate fitting of such an equation for different varieties may not be necessary. The one heat stress equation, the separate heat stress equation, and the Poikilotherm equation each had a better fit than nonlinear Briere et al. curves. The Poikilotherm equation showed promise, realistically capturing the high, low, and optimum rate values measured. All the equations showed promise to some degree for future applications in simulating the maize development rate. When fitting separate regressions for each variety for the heat stress equations, the base temperatures had a mean of 5.3 °C, similar to a previously published value of 6 °C. The last variety had noticeably different rates than the others. This study demonstrated that a simple approach (the heat stress equation) should be adequate in many cases. It also demonstrated that more detailed equations can be useful when a more mechanistic system is desired. Future research could investigate the reasons for the different development rate response of the last variety and investigate similar varieties.

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Oane, Mihai, Muhammad Arif Mahmood, and Andrei C. Popescu. "A State-of-the-Art Review on Integral Transform Technique in Laser–Material Interaction: Fourier and Non-Fourier Heat Equations." Materials 14, no. 16 (August 22, 2021): 4733. http://dx.doi.org/10.3390/ma14164733.

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Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties. Such wonderful features have enabled heat equations in various fields, including laser and electron beam processing. The integral transform technique (ITT) is a powerful general-purpose semi-analytical/numerical method that transforms partial differential equations into a coupled system of ordinary differential equations. Under this category, Fourier and non-Fourier heat equations can be implemented on both equilibrium and non-equilibrium thermo-dynamical processes, including a wide range of processes such as the Two-Temperature Model, ultra-fast laser irradiation, and biological processes. This review article focuses on heat equation models, including Fourier and non-Fourier heat equations. A comparison between Fourier and non-Fourier heat equations and their generalized solutions have been discussed. Various components of heat equations and their implementation in multiple processes have been illustrated. Besides, literature has been collected based on ITT implementation in various materials. Furthermore, a future outlook has been provided for Fourier and non-Fourier heat equations. It was found that the Fourier heat equation is simple to use but involves infinite speed heat propagation in comparison to the non-Fourier heat equation and can be linked with the Two-Temperature Model in a natural way. On the other hand, the non-Fourier heat equation is complex and involves various unknowns compared to the Fourier heat equation. Fourier and Non-Fourier heat equations have proved their reliability in the case of laser–metallic materials, electron beam–biological and –inorganic materials, laser–semiconducting materials, and laser–graphene material interactions. It has been identified that the material properties, electron–phonon relaxation time, and Eigen Values play an essential role in defining the precise results of Fourier and non-Fourier heat equations. In the case of laser–graphene interaction, a restriction has been identified from ITT. When computations are carried out for attosecond pulse durations, the laser wavelength approaches the nucleus-first electron separation distance, resulting in meaningless results.

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Tahir, M., G. Abbas, Kazuharu Bamba, and M. R. Shahzad. "Dynamics of dissipative self-gravitating source in Rastall gravity." International Journal of Modern Physics A 36, no. 20 (July 13, 2021): 2150153. http://dx.doi.org/10.1142/s0217751x21501530.

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The dynamics of dissipative gravitational collapse of a source is explored in Rastall gravity. The field equations are derived for the geometry and collapsing matter. The dynamical equations are formulated for the heat flux and diffusion approximation. The heat transportation equation is derived by using Müller–Israel–Stewart approach to investigate the effects of heat flux on the collapsing source. Moreover, an equation is found by combining the dynamical and heat transport equation, the consequences of this equation are discussed in detail. Furthermore, the Rastall parameter [Formula: see text] effect is analyzed for the collapse of sphere.

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Owino, Joseph Owuor. "GROUP ANALYSIS OF A NONLINEAR HEAT-LIKE EQUATION." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 11, no. 01 (January 13, 2023): 3113–31. http://dx.doi.org/10.47191/ijmcr/v11i1.03.

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We study a nonlinear heat like equation from a lie symmetry stand point. Heat equation have been employed to study ow of current, information and propagation of heat. The Lie group approach is used on the system to obtain symmetry reductions and the reduced systems studied for exact solutions. Solitary waves have been constructed by use of a linear span of time and space translation symmetries. We also compute conservation laws using multiplier approach and by a conservation theorem due to Ibragimov.

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Hills, Norman L., and John M. Irwin. "Infinite-Order Differential Equations and the Heat Equation." SIAM Journal on Mathematical Analysis 20, no. 2 (March 1989): 430–38. http://dx.doi.org/10.1137/0520029.

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Samdarshi, S. K., and S. C. Mullick. "Analytical Equation for the Top Heat Loss Factor of a Flat-Plate Collector With Double Glazing." Journal of Solar Energy Engineering 113, no. 2 (May 1, 1991): 117–22. http://dx.doi.org/10.1115/1.2929955.

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An analytical equation for the top heat loss factor of a flat-plate collector with double glazing has been developed. The maximum computational errors resulting from the use of this equation are plus or minus three percent compared to numerical solution of the heat balance equations. The equation is considerably more accurate than the currently used semi-empirical equations over the entire range of variables covered. It is found that the computational errors resulting from simplification of the proposed equation by approximation of the individual heat-transfer coefficients are much lower than the errors resulting from the use of semi-empirical equations.

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Gundogdu, Hami, and Ömer Gozukizil. "On the approximate numerical solutions of fractional heat equation with heat source and heat loss." Thermal Science, no. 00 (2021): 321. http://dx.doi.org/10.2298/tsci210713321g.

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In this paper, we are interested in obtaining an approximate numerical solution of the fractional heat equation where the fractional derivative is in Caputo sense. We also consider the heat equation with a heat source and heat loss. The fractional Laplace-Adomian decomposition method is applied to gain the approximate numerical solutions of these equations. We give the graphical representations of the solutions depending on the order of fractional derivatives. Maximum absolute error between the exact solutions and approximate solutions depending on the fractional-order are given. For the last thing, we draw a comparison between our results and found ones in the literature.

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Li, Jin. "Barycentric Rational Collocation Method for Nonlinear Heat Conduction Equation." Journal of Applied Mathematics 2022 (June 30, 2022): 1–9. http://dx.doi.org/10.1155/2022/8998193.

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Nonlinear heat equation solved by the barycentric rational collocation method (BRCM) is presented. Direct linearization method and Newton linearization method are presented to transform the nonlinear heat conduction equation into linear equations. The matrix form of nonlinear heat conduction equation is also obtained. Several numerical examples are provided to valid our schemes.

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Kafle, J., L. P. Bagale, and D. J. K. C. "Numerical Solution of Parabolic Partial Differential Equation by Using Finite Difference Method." Journal of Nepal Physical Society 6, no. 2 (December 31, 2020): 57–65. http://dx.doi.org/10.3126/jnphyssoc.v6i2.34858.

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In the real world, many physical problems like heat equation, wave equation, Laplace equation and Poisson equation are modeled by partial differential equations (PDEs). A PDE of the form ut = α uxx, (α > 0) where x and t are independent variables and u is a dependent variable; is a one-dimensional heat equation. This is an example of a prototypical parabolic equation. The heat equation has analytic solution in regular shape domain. If the domain has irregular shape, computing analytic solution of such equations is difficult. In this case, we can use numerical methods to compute the solution of such PDEs. Finite difference method is one of the numerical methods that is used to compute the solutions of PDEs by discretizing the given domain into finite number of regions. Here, we derived the Forward Time Central Space Scheme (FTCSS) for this heat equation. We also computed its numerical solution by using FTCSS. We compared the analytic solution and numerical solution for different homogeneous materials (for different values of diffusivity α). There is instantaneous heat transfer and heat loss for the materials with higher diffusivity (α) as compared to the materials of lower diffusivity. Finally, we compared simulation results of different non-homogeneous materials.

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Hsiao, Kai-Long, and Guan-Bang Chen. "Conjugate Heat Transfer of Mixed Convection for Viscoelastic Fluid Past a Stretching Sheet." Mathematical Problems in Engineering 2007 (2007): 1–21. http://dx.doi.org/10.1155/2007/17058.

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A conjugate heat transfer problem of a second-grade viscoelastic fluid past a stretching sheet has been studied. Governing equations include heat conduction equation of a stretching sheet, continuity equation, momentum equation, and energy equation of a second-grade fluid, analyzed by a combination of a series expansion method, the similarity transformation, and a second-order accurate finite-difference method. These solutions are used to iterate with the heat conduction equation of the stretching sheet to obtain distributions of the local convective heat transfer coefficient and the stretching sheet temperature. Ranges of dimensionless parameters, the Prandtl numberPr, the elastic numberEand the conduction-convection coefficientNccare from 0.001 to 10, 0.0001 to 0.01, and 0.5 to 2.0, respectively. A parameterG, which is used to represent the dominance of the buoyant effect, is present in governing equations. Results indicated that elastic effect in the flow could increase the local heat transfer coefficient and enhance the heat transfer of a stretching sheet. In addition, same as the results from Newtonian fluid flow and conduction analysis of a stretching sheet, a better heat transfer is obtained with a largerNcc,G, andE.

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Farg, Ahmed Saeed, A. M. Abd Elbary, and Tarek A. Khalil. "Applied method of characteristics on 2^nd order linear P.D.E." Journal of Physics: Conference Series 2304, no. 1 (August 1, 2022): 012003. http://dx.doi.org/10.1088/1742-6596/2304/1/012003.

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Abstract PDEs are very important in dynamics, aerodynamics, elasticity, heat transfer, waves, electromagnetic theory, transmission lines, quantum mechanics, weather forecasting, prediction of crime places, disasters, how universe behave ……. Etc., second order linear PDEs can be classified according to the characteristic equation into 3 types hyperbolic, parabolic and elliptic; Hyperbolic equations have two distinct families of (real) characteristic curves, parabolic equations have a single family of characteristic curves, and the elliptic equations have none. All the three types of equations can be reduced to its first canonical form finding the general solution or the second canonical form similar to 3 basic PDE models; Hyperbolic equations reduce to a form coinciding with the wave equation in the leading terms, the parabolic equations reduce to a form modeled by the heat equation, and the Laplace’s equation models the canonical form of elliptic equations. Thus, the wave, heat and Laplace’s equations serve as basic canonical models for all second order linear PDEs.

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Kebeba, Nimona Ketema, and Gizaw Debito Haifo. "Analysis of Two-Dimensional Heat Transfer Problem Using the Boundary Integral Equation." Advances in Mathematical Physics 2022 (November 14, 2022): 1–7. http://dx.doi.org/10.1155/2022/1889774.

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In this paper, we examine the problem of two-dimensional heat equations with certain initial and boundary conditions being considered. In a two-dimensional heat transport problem, the boundary integral equation technique was applied. The problem is expressed by an integral equation using the fundamental solution in Green’s identity. In this study, we transform the boundary value problem for the steady-state heat transfer problem into a boundary integral equation and drive the solution of the two-dimensional heat transfer problem using the boundary integral equation for the mixed boundary value problem by using Green’s identity and fundamental solution.

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Jameel, A. F. "Semi-Analytical Solution of Heat Equation in Fuzzy Environment." International Journal of Applied Physics and Mathematics 4, no. 6 (2014): 371–78. http://dx.doi.org/10.17706/ijapm.2014.4.6.371-378.

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ABATANGELO, Vito, and Sorin DRAGOMIR. "Discrete Heat Equation Morphisms." Interdisciplinary Information Sciences 14, no. 2 (2008): 225–44. http://dx.doi.org/10.4036/iis.2008.225.

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Hayashi, Nakao, Elena I. Kaikina, and Pavel I. Naumkin. "Subcritical nonlinear heat equation." Journal of Differential Equations 238, no. 2 (July 2007): 366–80. http://dx.doi.org/10.1016/j.jde.2007.04.007.

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Shishkina, E. L., and A. K. Yusupova. "On Singular Heat Equation." Differential Equations 59, no. 12 (December 2023): 1708–20. http://dx.doi.org/10.1134/s001226612312011x.

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Aghili, Arman. "Non-homogeneous impulsive time fractional heat conduction equation." Journal of Numerical Analysis and Approximation Theory 52, no. 1 (July 10, 2023): 22–33. http://dx.doi.org/10.33993/jnaat521-1316.

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This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.

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Gomes, Michelle G., Nattácia R. A. F. Rocha, Alex A. Moura, Nadine P. Merlo, Moilton R. Franco Júnior, and Patrisia O. Rodrigues. "Prediction of Liquid Molar Volume and Heat of Vaporization of Fatty Acids Using an Equation of State." Current Physical Chemistry 10, no. 3 (November 4, 2020): 189–98. http://dx.doi.org/10.2174/1877946809666191129110018.

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Background:: The liquid molar volume (V) and the heat of vaporization (ΔHVAP) of four fatty acids (n-Heptanoic acid, Hexadecanoic acid, n-Hexanoic acid and n- Dodecanoic acid) have been estimated. Objective:: This paper aims to calculate the liquid molar volume and the heat of vaporization of four fatty acids under the critical point using two traditional equations of state: Peng-Robinson (PR) [21] and Soave-Redlich-Kwong. Methods: The area rules method applicable to obtaining the saturation pressure of the compounds has been used. The properties of the acids investigated in this work have been compared with those provided by literature. For molar volumes, the equations of state have given improved predictions when compared to traditional equations such as Rackett equation and so on. According to the vapor enthalpy calculations, no reference value was required. Results: In general, the Clausius-Clapeyron equation provides a better estimation of the vaporization enthalpy of fatty acids when Soave-Redlich-Kwong (SRK) equation was used. The heat of vaporization for fatty acids can be calculated with good reliability in comparison with the Watson equation if suitable equation of state is used. Conclusion: Accurate results for heat of vaporization can be reached in comparison with the Watson equation if the reliable equation of state is used.

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Zhong, Wan Li, Wei Wang, Jie Dong Lin, Ming Nie, and Chang Hong Liu. "Rough Set Analysis of the Heat Conduction in the Steam Pipe." Applied Mechanics and Materials 750 (April 2015): 371–75. http://dx.doi.org/10.4028/www.scientific.net/amm.750.371.

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During the analysis of stability heat conduction in the composite pipes, firstly, when the heat equation contained fuzzy and random uncertain parameters, interval equations of the heat conduction are presented in the rough set. Secondly, the error expecting of heat conduction equation is presented. Finally, with upper (lower) approximation in rough set, a new method of the rough set analysis to deal with the stability heat conduction is presented.

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Kosmakova, M. T., S. A. Iskakov, and L. Zh Kasymova. "To solving the fractionally loaded heat equation." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 101, no. 1 (March 30, 2021): 65–77. http://dx.doi.org/10.31489/2021m1/65-77.

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In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions. Research methods are based on an approach to the study of boundary value problems, based on their reduction to integral equations. The problem is reduced to a Volterra integral equation of the second kind by inverting the differential part. We also carried out a study the limit cases for the fractional derivative order of the term with a load in the heat equation of the boundary value problem. It is shown that the existence and uniqueness of solutions to the integral equation depends on the order of the fractional derivative in the loaded term.

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Khalaf, Maki H., and Manar S. Mahdi. "A Theoretical Study on Mixed Convection in a Horizontal Annulus." Tikrit Journal of Engineering Sciences 13, no. 3 (October 1, 2006): 62–87. http://dx.doi.org/10.25130/tjes.13.3.10.

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A theoretical study has been conducted on mixed convection heat transfer of the flow through a horizontal annulus the outer surface heated with an axial uniform heat flux while the inner surface cooled at constant surface temperature. Theoretically the governing equations for a flow were reduced to four equations, which are continuity equation, radial and tangential momentum equation, axial momentum equation and vorticity equation in which the variables were the temperature, vorticity, stream function and axial velocity. These equations were reduced to dimensionless equations in which Reynolds, Prandtl and Rayleigh numbers were presented. These equations were solved numerically by using the marching process explicit finite difference method and Gauss elimination technique after changing the elliptic type energy and momentum equations to parabolic form by adding the change with time for each variable to the left hand side of these equations. Numerical results for the annuli heated by a uniform heat flux in the fully developed region were obtained and represented by stream function contours and isotherms for different values of Rayleigh and circumferential distribution of local Nussult number. The results were based on the fact that the secondary flow created by natural convection has significant effects on the heat transfer process, and reveal an increase in the Nussult number values as the heat flux increases in the horizontal position.

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Saanouni, Tarek. "Global Well-Posedness and Finite-Time Blow-Up of Some Heat-Type Equations." Proceedings of the Edinburgh Mathematical Society 60, no. 2 (November 2, 2016): 481–97. http://dx.doi.org/10.1017/s0013091516000213.

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AbstractWe study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

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Baker-Jarvis, J., and R. Inguva. "Heat Conduction in Heterogeneous Materials." Journal of Heat Transfer 107, no. 1 (February 1, 1985): 39–43. http://dx.doi.org/10.1115/1.3247399.

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A new solution to the heat equation in composite media is derived using a variational principle developed by Ben-Amoz. The model microstructure is fed into the equations via a term for the polar moment of the inclusions in a representative volume. The general solution is presented as an integral in terms of sources and a Green function. The problem of uniqueness is studied to determine appropriate boundary conditions. The solution reduces to the solution of the normal heat equation in the limit of homogeneous media.

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Xiao, Xufeng, Dongwei Gui, and Xinlong Feng. "A highly efficient operator-splitting finite element method for 2D/3D nonlinear Allen–Cahn equation." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 2 (February 6, 2017): 530–42. http://dx.doi.org/10.1108/hff-12-2015-0521.

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Purpose This study aims to present a highly efficient operator-splitting finite element method for the nonlinear two-dimensional/three-dimensional (2D/3D) Allen–Cahn (AC) equation which describes the anti-phase domain coarsening in a binary alloy. This method is presented to overcome the higher storage requirements, computational complexity and the nonlinear term in numerical computation for the 2D/3D AC equation. Design/methodology/approach In each time interval, the authors first split the original equation into a heat equation and a nonlinear equation. Then, they split the high-dimensional heat equation into a series of one-dimensional (1D) heat equations. By solving each 1D subproblem, the authors obtain a numerical solution for heat equation and take it as an initial for the nonlinear equation, which is solved analytically. Findings The authors show that the proposed method is unconditionally stable. Finally, various numerical experiments are presented to confirm the high accuracy and efficiency of this method. Originality/value A new operator-splitting method is presented for solving the 2D/3D parabolic equation. The 2D/3D parabolic equation is split into a sequence of 1D parabolic equations. In comparison with standard finite element method, the present method can save much central processing unit time. Stability analysis and error estimates are derived and numerical results are presented to support the theoretical analysis.

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Neves, Aloisio F. "On the strongly damped wave equation and the heat equation with mixed boundary conditions." Abstract and Applied Analysis 5, no. 3 (2000): 175–89. http://dx.doi.org/10.1155/s1085337500000348.

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We study two one-dimensional equations: the strongly damped wave equation and the heat equation, both with mixed boundary conditions. We prove the existence of global strong solutions and the existence of compact global attractors for these equations in two different spaces.

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Chen, Gang. "Ballistic-Diffusive Equations for Transient Heat Conduction From Nano to Macroscales." Journal of Heat Transfer 124, no. 2 (August 6, 2001): 320–28. http://dx.doi.org/10.1115/1.1447938.

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In this paper, we present the derivation of a new type of heat conduction equations, named as the ballistic-diffusive equations, that are suitable for describing transient heat conduction from nano to macroscale. The equations are derived from the Boltzmann equation under the relaxation time approximation. The distribution function is divided into two parts. One represents the ballistic transport originating from the boundaries and the other is the transport of the scattered and excited carriers. The latter is further approximated as a diffusive process. The obtained ballistic-diffusive equations are applied to the problem of transient heat conduction across a thin film and the results are compared to the solutions for the same problem based on the Boltzmann equation, the Fourier law, and the Cattaneo equation. The comparison suggests that the ballistic-diffusive equations can be a useful tool in dealing with transient heat conduction problems from nano to macroscales. Boundary conditions for the derived equations are also discussed. Special emphasis is placed on the consistency of temperature used in the boundary conditions and in the equations.

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Glagolev, M. V., E. A. Dyukarev, I. E. Terentieva, and A. F. Sabrekov. "On a question of non-constant thermal diffusivity of soils." IOP Conference Series: Earth and Environmental Science 1093, no. 1 (September 1, 2022): 012019. http://dx.doi.org/10.1088/1755-1315/1093/1/012019.

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Abstract The general heat conductivity equation includes time- and depth-dependent soil properties (soil heat capacity and thermal conductivity). The simplified form of the heat conductivity equation contains only the soil thermal diffusivity parameter. Numerical solutions of the general and simplified equations were compared to quantify the possibility of equation reduction. Two test runs for soils with different compositions were done. The thermal regime for both peat soil and dark chestnut soil does not change significantly after using a simplified heat equation according to model estimations. The maximal soil temperature discrepancy was about 0.5 °C for peat soil and 2.2-3.3 °C for dark chestnut soil, which results in 4-6% error in methane efflux estimations.

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32

Samdarshi, S. K., and S. C. Mullick. "Generalized Analytical Equation for the Top Heat Loss Factor of a Flat-Plate Solar Collector With N Glass Covers." Journal of Solar Energy Engineering 116, no. 1 (February 1, 1994): 43–46. http://dx.doi.org/10.1115/1.2930064.

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A generalized analytical equation for the top heat loss factor of a flat-plate collector with one or more glass covers has been developed. The maximum computational errors resulting from the use of the analytical equation with several simplifications are ± 5 percent compared to numerical solution of the set of heat balance equations. The analytical equation is considerably more accurate than the available semi-empirical equations over the entire range of variables covered. An additional advantage of the proposed technique over the semi-empirical equations is that results can be obtained for different values of sky temperature, using any given correlation for convective heat transfer in the air gap spacings, and for any given values of fluid (air in the present case) properties.

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33

Vadakkan, Unnikrishnan, Suresh V. Garimella, and Jayathi Y. Murthy. "Transport in Flat Heat Pipes at High Heat Fluxes From Multiple Discrete Sources." Journal of Heat Transfer 126, no. 3 (June 1, 2004): 347–54. http://dx.doi.org/10.1115/1.1737773.

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A three-dimensional model has been developed to analyze the transient and steady-state performance of flat heat pipes subjected to heating with multiple discrete heat sources. Three-dimensional flow and energy equations are solved in the vapor and liquid regions, along with conduction in the wall. Saturated flow models are used for heat transfer and fluid flow through the wick. In the wick region, the analysis uses an equilibrium model for heat transfer and a Brinkman-Forchheimer extended Darcy model for fluid flow. Averaged properties weighted with the porosity are used for the wick analysis. The state equation is used in the vapor core to relate density change to the operating pressure. The density change due to pressurization of the vapor core is accounted for in the continuity equation. Vapor flow, temperature and hydrodynamic pressure fields are computed at each time step from coupled continuity/momentum and energy equations in the wick and vapor regions. The mass flow rate at the interface is obtained from the application of kinetic theory. Predictions are made for the magnitude of heat flux at which dryout would occur in a flat heat pipe. The input heat flux and the spacing between the discrete heat sources are studied as parameters. The location in the heat pipe at which dryout is initiated is found to be different from that of the maximum temperature. The location where the maximum capillary pressure head is realized also changes during the transient. Axial conduction through the wall and wick are seen to play a significant role in determining the axial temperature variation.

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Font, Francesc. "Memory and nonlocal effects of heat transport in a spherical nanoparticle." Journal of Physics: Conference Series 2116, no. 1 (November 1, 2021): 012055. http://dx.doi.org/10.1088/1742-6596/2116/1/012055.

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Abstract In this paper a mathematical model describing the heat transport in a spherical nanoparticle subject to Newton heating at its surface is presented. The governing equations involve a phonon hydrodynamic equation for the heat flux and the classical energy equation that relates the heat flux and the temperature. Assuming radial symmetry the model is reduced to two partial differential equation, one for the radial component of the flux and one for the temperature. We solve the model numerically by means of finite differences. The resulting temperature profiles show characteristic wave-like behaviour consistent with the non Fourier components in the hydrodynamic equation.

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35

Al-Mahdawi, Hassan K. Ibrahim, Mostafa Abotaleb, Hussein Alkattan, Al-Mahdawi Zena Tareq, Amr Badr, and Ammar Kadi. "Multigrid Method for Solving Inverse Problems for Heat Equation." Mathematics 10, no. 15 (August 7, 2022): 2802. http://dx.doi.org/10.3390/math10152802.

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In this paper, the inverse problems for the boundary value and initial value in a heat equation are posed and solved. It is well known that those problems are ill posed. The problems are reformulated as integral equations of the first kind by using the separation-of-variables method. The discretization of the integral equation allowed us to reduce the integral equation to a system of linear algebraic equations or a linear operator equation of the first kind on Hilbert spaces. The Landweber-type iterative method was used in order to find an approximation solution. The V-cycle multigrid method is used to obtain more frequent and fast convergence for iteration. The numerical computation examples are presented to verify the accuracy and fast computing of the approximation solution.

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36

Komar, Ludmila. "Temperature Effects in Polymer Materials Numerical Solution with Using Material Process by a Pulsed Ion Beams Taking Into Account the Heat Flows Relaxation Parameter." Вестник Пермского университета. Математика. Механика. Информатика, no. 3(58) (2022): 38–48. http://dx.doi.org/10.17072/1993-0550-2022-3-38-48.

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A one-dimensional problem numerical simulation of the thermal wave motion in a material is carried out. The material surface is processed by a pulsed ion beam. The pulse action as a source function is given by a linear dependence on a time and an ion penetration depth into the material. The above heat-conduction equation is a non-linear differential equation with a parameter which presents the relaxation time of the heat flux. The heat-conduction equation is supplemented by the heat flow change in time equation, which was proposed by Cattaneo and Vernotte. The temperature and heat flux equations system is numerically solved with using the finite difference method. A significant relaxation parameter value influence on the temperature profiles formation in the treated material surface vicinity is shown.

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K.C., Durga Jang, and Ganesh Bahadur Basnet. "Discrete Maximum Principle in One-Dimensional Heat Equation." Journal of Advanced College of Engineering and Management 2 (November 29, 2016): 5. http://dx.doi.org/10.3126/jacem.v2i0.16093.

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The maximum principle plays key role in the theory and application of a wide class of real linear partial differential equations. In this paper, we introduce ‘Maximum principle and its discrete version’ for the study of second-order parabolic equations, especially for the one-dimensional heat equation. We also give a short introduction of formation of grid as well as finite difference schemes and a short prove of the ‘Discrete Maximum principle’ by using different schemes of heat equation.Journal of Advanced College of Engineering and Management, Vol. 2, 2016, Page: 5-10

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38

Shah, Rehan Ali, Hidayat Ullah, Muhammad Sohail Khan, and Aamir Khan. "Parametric analysis of the heat transfer behavior of the nano-particle ionic-liquid flow between concentric cylinders." Advances in Mechanical Engineering 13, no. 6 (June 2021): 168781402110240. http://dx.doi.org/10.1177/16878140211024009.

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This paper investigates the enhanced viscous behavior and heat transfer phenomenon of an unsteady two di-mensional, incompressible ionic-nano-liquid squeezing flow between two infinite parallel concentric cylinders. To analyze heat transfer ability, three different type nanoparticles such as Copper, Aluminum [Formula: see text], and Titanium oxide [Formula: see text] of volume fraction ranging from 0.1 to 0.7 nm, are added to the ionic liquid in turns. The Brinkman model of viscosity and Maxwell-Garnets model of thermal conductivity for nano particles are adopted. Further, Heat source [Formula: see text], is applied between the concentric cylinders. The physical phenomenon is transformed into a system of partial differential equations by modified Navier-Stokes equation, Poisson equation, Nernst-Plank equation, and energy equation. The system of nonlinear partial differential equations, is converted to a system of coupled ordinary differential equations by opting suitable transformations. Solution of the system of coupled ordinary differential equations is carried out by parametric continuation (PC) and BVP4c matlab based numerical methods. Effects of squeeze number ( S), volume fraction [Formula: see text], Prandtle number (Pr), Schmidt number [Formula: see text], and heat source [Formula: see text] on nano-ionicliquid flow, ions concentration distribution, heat transfer rate and other physical quantities of interest are tabulated, graphed, and discussed. It is found that [Formula: see text] and Cu as nanosolid, show almost the same enhancement in heat transfer rate for Pr = 0.2, 0.4, 0.6.

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Afify, Ahmed A., and Nasser S. Elgazery. "Lie group analysis for the effects of chemical reaction on MHD stagnation-point flow of heat and mass transfer towards a heated porous stretching sheet with suction or injection." Nonlinear Analysis: Modelling and Control 17, no. 1 (January 25, 2012): 1–15. http://dx.doi.org/10.15388/na.17.1.14074.

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An analysis is carried out to study two dimensional stagnation-point flow of heat and mass transfer of an incompressible, electrically conducting fluid towards a heated porous stretching sheet embedded in a porous medium in the presence of chemical reaction, heat generation/absorption and suction or injection effects. A scaling group of transformations is applied to the governing equations. After finding three absolute invariants a third order ordinary differential equation corresponding to the momentum equation and two second order ordinary differential equation corresponding to energy and diffusion equations are derived. Furthermore the similarity equations are solved numerically by using shooting technique with fourth-order Runge–Kutta integration scheme. A comparison with known results is excellent. The phenomenon of stagnation-point flow towards a heated porous stretching sheet in the presence of chemical reaction, suction or injection with heat generation/absorption effects play an important role on MHD heat and mass transfer boundary layer. The results thus obtained are presented graphically and discussed.

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Singh, Khilap, and Manoj Kumar. "Influence of Chemical Reaction on Heat and Mass Transfer Flow of a Micropolar Fluid over a Permeable Channel with Radiation and Heat Generation." Journal of Thermodynamics 2016 (December 7, 2016): 1–10. http://dx.doi.org/10.1155/2016/8307980.

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The effects of chemical reaction on heat and mass transfer flow of a micropolar fluid in a permeable channel with heat generation and thermal radiation is studied. The Rosseland approximations are used to describe the radiative heat flux in the energy equation. The model contains nonlinear coupled partial differential equations which have been transformed into ordinary differential equation by using the similarity variables. The relevant nonlinear equations have been solved by Runge-Kutta-Fehlberg fourth fifth-order method with shooting technique. The physical significance of interesting parameters on the flow and heat transfer characteristics as well as the local skin friction coefficient, wall couple stress, and the heat transfer rate are thoroughly examined.

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41

Abdullayev, Akmaljon, Kholsaid Kholturayev, and Nigora Safarbayeva. "Exact method to solve of linear heat transfer problems." E3S Web of Conferences 264 (2021): 02059. http://dx.doi.org/10.1051/e3sconf/202126402059.

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When approximating multidimensional partial differential equations, the values of the grid functions from neighboring layers are taken from the previous time layer or approximation. As a result, along with the approximation discrepancy, an additional discrepancy of the numerical solution is formed. To reduce this discrepancy when solving a stationary elliptic equation, parabolization is carried out, and the resulting equation is solved by the method of successive approximations. This discrepancy is eliminated in the approximate analytical method proposed below for solving two-dimensional equations of parabolic and elliptic types, and an exact solution of the system of finite difference equations for a fixed time is obtained. To solve problems with a boundary condition of the first kind on the first coordinate and arbitrary combinations of the first, second and third kinds of boundary conditions on the second coordinate, it is proposed to use the method of straight lines on the first coordinate and ordinary sweep method on the second coordinate. Approximating the equations on the first coordinate, a matrix equation is built relative to the grid functions. Using eigenvalues and vectors of the three-diagonal transition matrix, linear combinations of grid functions are compiled, where the coefficients are the elements of the eigenvectors of the three-diagonal transition matrix. Boundary conditions, and for a parabolic equation, initial conditions are formed for a given combination of grid functions. The resulting one-dimensional differential-difference equations are solved by the ordinary sweep method. From the resulting solution, proceed to the initial grid functions. The method provides a second order of approximation accuracy on coordinates. And the approximation accuracy in time when solving the parabolic equation can be increased to the second order using the central difference in time. The method is used to solve heat transfer problems when the boundary conditions are expressed by smooth and discontinuous functions of a stationary and non-stationary nature, and the right-hand side of the equation represents a moving source or outflow of heat.

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42

Papanicolaou, Vassilis G., Eva Kallitsi, and George Smyrlis. "Entire solutions for the heat equation." Electronic Journal of Differential Equations 2021, no. 01-104 (May 25, 2021): 4. http://dx.doi.org/10.58997/ejde.2021.44.

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We consider the solutions of the heat equation $$ \partial_t F = \partial_z^2 F $$ which are entire in z and t (caloric functions). We examine the relation of the z-order and z-type of an entire caloric function $F(t, z)$, viewed as function of z, to its t-order and t-type respectively, if it is viewed as function of $t$. Also, regarding the zeros $z_k(t) $ of an entire caloric function $F(t, z)$, viewed as function of $z$, we show that the points $(t, z) $ at which $$ F(t, z) = \partial_z F(t, z) = 0 $$ form a discrete set in $\mathbb{C}^2$ and, then, we derive the t-evolution equations of $z_k(t) $. These are differential equations that hold for all but countably many ts in $\mathbb{C}$. For more information see https://ejde.math.txstate.edu/Volumes/2021/44/abstr.html

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43

Zak, Michail. "Modelling ‘Life’ against ‘heat death’." International Journal of Astrobiology 17, no. 1 (April 24, 2017): 61–69. http://dx.doi.org/10.1017/s147355041700009x.

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AbstractThis work is inspired by the discovery of a new class of dynamical system described by ordinary differential equations coupled with their Liouville equation. These systems called self-controlled since the role of actuators is played by the probability produced by the Liouville equation. Following the Madelung equation that belongs to this class, non-Newtonian properties such as randomness, entanglement and probability interference typical for quantum systems have been described. Special attention was paid to the capability to violate the second law of thermodynamics, which makes these systems neither Newtonian, nor quantum. It has been shown that self-controlled dynamical systems can be linked to mathematical models of living systems. The discovery of isolated dynamical systems that can decrease entropy in violation of the second law of thermodynamics, and resemblances of these systems to livings suggests that ‘Life’ can slow down the ‘heat death’ of the Universe and that can be associated with the Purpose of Life.

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44

Yesbayev, A. N. "INVESTIGATION OF THE MODEL FOR THE ESSENTIALLY LOADED HEAT EQUATION." Eurasian Physical Technical Journal 16, no. 1 (June 14, 2019): 121–28. http://dx.doi.org/10.31489/2019no1/121-128.

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45

Saleh, M. M., I. L. El-Kalla, and M. M. Ehab. "Stochastic Finite Element Technique for Stochastic One-Dimension Time-Dependent Differential Equations with Random Coefficients." Differential Equations and Nonlinear Mechanics 2007 (2007): 1–16. http://dx.doi.org/10.1155/2007/48527.

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The stochastic finite element method (SFEM) is employed for solving stochastic one-dimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.

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Shi, Wei, and Li Li Ma. "Concise BEM for State Heat Conduction of FGMs." Advanced Materials Research 308-310 (August 2011): 473–76. http://dx.doi.org/10.4028/www.scientific.net/amr.308-310.473.

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The thermal conductivity problem of functionally graded plate is studied under different temperature fields by a new concise BEM in this paper. At first, we convert the heat conduction differential equation of functionally graded materials (FGMs) to a homogeneous material thermal conductivity equation by using variable substitution, then the Galerkin boundary integral equation is reduced to be a system of linear equations. Finally we arrive at an internal temperature of objects, and plot the distribution graphics and effects of material parameters on temperature distribution. It proves that the new concise BEM is very effective.

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Mulyati, Annisa Eki, and Sugiyanto Sugiyanto. "Aplikasi Persamaan Bessel Orde Nol Pada Persamaan Panas Dua Dimensi." Jurnal Fourier 2, no. 2 (October 31, 2013): 113. http://dx.doi.org/10.14421/fourier.2013.22.113-123.

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Bessel differential equation is one of the applied equation in physics is about heat transfer. Application of modified Bessel function of order zero on heat transfer process of two-dimensional objects which can be modelled in the form of a two-order partial differential equations as follows, ..... With the obtained solutions of Bessel's differential equation application of circular fin, .... two-dimensional temperature stated on the point ..... against time t

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Sedki, Ahmed M., S. M. Abo-Dahab, J. Bouslimi, and K. H. Mahmoud. "Thermal radiation effect on unsteady mixed convection boundary layer flow and heat transfer of nanofluid over permeable stretching surface through porous medium in the presence of heat generation." Science Progress 104, no. 3 (July 2021): 003685042110422. http://dx.doi.org/10.1177/00368504211042261.

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Here, we study the effect of mixed convection and thermal radiation on unsteady boundary layer of heat transfer and nanofluid flow over permeable moving surface through a porous medium. The effect of heat generation is also discussed. The equations governing the system are the continuity equation, momentum equation and the heat transfer equation. These governing equations transformed into a system of nondimensional equations contain many physical parameters that describe the study. The transformed equations are solved numerically using an implicit finite difference technique with Newton's linearization method. The thermo-physical parameters describe the study are the mixed convection parameter α, [Formula: see text], the Radiation parameter Rd, [Formula: see text] , porous medium parameter k, [Formula: see text], the nanoparticles volume [Formula: see text],[Formula: see text], the suction or injection parameter fw, [Formula: see text], the unsteadiness parameter At, [Formula: see text] and the heat source parameter λ = 0.5 .The influence of the thermo-physical parameters is obtained analytically and displayed graphically. Comparisons of some special cases of the present study are performed with previously published studies and a good agreement is obtained.

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Burma Saparova, Roza Mamytova, Nurjamal Kurbanbaeva, and Anvarjon Akhatjonovich Ahmedov. "A Haar Wavelet Series Solution of Heat Equation with Involution." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 86, no. 2 (August 22, 2021): 50–55. http://dx.doi.org/10.37934/arfmts.86.2.5055.

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It is well known that the wavelets have widely applied to solve mathematical problems connected with the differential and integral equations. The application of the wavelets possess several important properties, such as orthogonality, compact support, exact representation of polynomials at certain degree and the ability to represent functions on different levels of resolution. In this paper, new methods based on wavelet expansion are considered to solve problems arising in approximation of the solution of heat equation with involution. We have developed new numerical techniques to solve heat equation with involution and obtained new approximative representation for solution of heat equations.

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Murphy, Conor N., Luísa Toledo Tude, and Paul R. Eastham. "Laser Cooling beyond Rate Equations: Approaches from Quantum Thermodynamics." Applied Sciences 12, no. 3 (February 3, 2022): 1620. http://dx.doi.org/10.3390/app12031620.

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Solids can be cooled by driving impurity ions with lasers, allowing them to transfer heat from the lattice phonons to the electromagnetic surroundings. This exemplifies a quantum thermal machine, which uses a quantum system as a working medium to transfer heat between reservoirs. We review the derivation of the Bloch-Redfield equation for a quantum system coupled to a reservoir, and its extension, using counting fields, to calculate heat currents. We use the full form of this equation, which makes only the weak-coupling and Markovian approximations, to calculate the cooling power for a simple model of laser cooling. We compare its predictions with two other time-local master equations: the secular approximation to the full Bloch-Redfield equation, and the Lindblad form expected for phonon transitions in the absence of driving. We conclude that the full Bloch-Redfield equation provides accurate results for the heat current in both the weak- and strong- driving regimes, whereas the other forms have more limited applicability. Our results support the use of Bloch-Redfield equations in quantum thermal machines, despite their potential to give unphysical results.

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Journal articles on the topic 'Heat equation'

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