Solution Of Heat Equation Research Articles

A Haar Wavelet Series Solution of Heat Equation with Involution

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.

Analytical temperature profiles for laser-irradiated gold nanorod immersed in an optically non-absorbing medium

Laser heating of gold nanorods immersed in an optically non-absorbing medium is important in both therapeutic and diagnostic medical procedures. For both continuous wave and pulsed lasers, we derive analytical temperature solutions by modeling a nanorod as a prolate spheroid. For the continuous wave laser irradiation, an exact analytical solution for the steady state heat transfer case using integer-degree Legendre function expansions in prolate spheroidal coordinates is presented. The steady state solution reveals that for the case of gold nanorod immersed in water, there is insignificant temperature variations on any single spheroidal surface that is coincident or confocal to the prolate spheroid surface, and the temperature only changes across spheroidal surfaces within the immediate surrounding water. This is found to be due to the very high thermal conductivity of gold which is more than a hundred-fold higher than that of water. The insight that the temperature rise is nearly invariant on any single spheroidal surface, along with a dimensional analysis, helps reduce the complexity of the heat equation in the unsteady regime of a pulsed laser illumination, and leads to a simple, lower dimensional model of unsteady heat equation, which takes into account the temporal dependency and spatial dependency in the direction across spheroidal surfaces. For the reduced unsteady heat equation, a simple, efficient, and insightful solution algorithm based on the Laplace transform and complex-degree Legendre functions is proposed. The maximum temperature and temperature profiles as functions of laser power, and nanorod geometries are obtained, revealing that for a fixed laser fluence and fixed nanorod mass, the more stretched nanorods yield higher temperatures in the spheroid and its immediate surroundings. In other words, more stretched nanorods are more efficient in converting the laser energy into localized, elevated temperatures. The obtained temperature solutions are useful in designing a parameter space for optimum therapeutic procedures for ablating cancer cells using high temperatures generated in the immediate vicinity of nanoparticles. The unsteady temperature solutions can also be used to obtain solutions to thermo-elastic models which are used to compute the photoacoustic signals that can, in turn, be used for cellular imaging and diagnostics.

Physics‐based numerical simulation of heat and moisture transport in red chili subjected to mix‐mode solar drying with phase change energy storage

AbstractIn this study, a coupled heat and mass transfer model was developed using a finite element (FE) simulation approach to describe solar drying of red chili integrated with Na2SO4∙10H2O as phase change material thermal storage. The coupled heat and mass transfer equation solution was achieved by using COMSOL multiphysics with three‐dimensional (3D) axisymmetric geometry mesh generation. The Lagrange triangular FEs of very fine size were employed in obtaining the geometry of the mesh. This study adopts a time‐dependent study that shows the variation of time within 0, 2, 3, 4, and 5 hours, respectively, for continuous drying of red chili in the solar dryer. The effect of air temperature and thermal storage on temperature and moisture distribution on drying of red chili was investigated. The temperature distribution was not uniform and heat transfer at the base was higher than the tip while there is a clear moisture gradient for different drying times, though this moisture moves from the base to the tip and more uniform distribution was obtained after 4 hours of drying. Drying temperature increased from a side to the core of the sample as the drying progresses. Although thermal storage accelerated the drying process, solar radiation showed a greater influence on the drying process. The developed numerical model with the experimental results showed the same trend with the values of MAE and SE for temperature found to be 0.48132 and 0.000842 for the without thermal storage (NTS) sample, 0.2266 and 30.000551 for thermal storage (TS). Also, the values of MAE and SE are 0.0051 and 0.006048 for NTS, 0.00212 and 0.00538 for TS, respectively for moisture profile. The lower values of these errors specify a better prediction capability of the model code.

An Approximate Solution of the Space Fractional-Order Heat Equation by the Non-Polynomial Spline Functions

The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear non-polynomial spline to a two-dimensional spline to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15.

APPLICATION OF NEW ITERATIVE ALGORITHM FOR SOLVING NONLINEAR CONVECTION-DIFFUSION HEAT EQUATION WITH CONSTANT COEFFICIENTS

This paper presents computational procedures to formulate an algorithm based on the new iterative method (NIM) for the numerical solution of nonlinear convection-diffusion heat equation with constant coefficients. The newly formulated algorithm (NIA) was fully described the relationship between convection and diffusion constants. Three test cases (prototype) are consider to investigate the time distribution profiles in heat equation other study. The algorithm is easy and efficient to solve similar problems in physical sciences.

Optimal local truncation error method for solution of wave and heat equations for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

Recently we have developed the optimal local truncation error method (OLTEM) for PDEs with constant coefficients on irregular domains and unfitted Cartesian meshes. However, many important engineering applications include domains with different material properties (e.g., different inclusions, multi-material structural components, etc.) for which this technique cannot be directly applied. In the paper OLTEM is extended to a much more general case of PDEs with discontinuous coefficients and can treat the above-mentioned applications. We show the development of OLTEM for the 1-D and 2-D scalar wave equation as well as the heat equation using compact 3-point (in the 1-D case) and 9-point (in the 2-D case) stencils that are similar to those for linear quadrilateral finite elements. Trivial unfitted Cartesian meshes are used for OLTEM with complex interfaces between different materials. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at a given width of stencil equations. In contrast to the second order of accuracy for linear finite elements, OLTEM provides the fourth order of accuracy in the 1-D case and in the 2-D case for horizontal interfaces as well as the third order of accuracy for the general geometry of smooth interfaces. The numerical results for the domains with complex smooth interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than quadratic finite elements and yields the results close to those for cubic finite elements with much wider stencils. The wave and heat equations are uniformly treated with OLTEM. OLTEM can be directly applied to other partial differential equations.

Sample paths of white noise in spaces with dominating mixed smoothness

The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smoothness in tangential, normal and time direction.

Asymptotically self-similar global solutions for a complex-valued quadratic heat equation with a generalized kernel

Asymptotically self-similar global solutions for a complex-valued quadratic heat equation with a generalized kernel

Solution of heat equation (Partial Differential Equation) by various methods

In the present paper we solved heat equation (Partial Differential Equation) by various methods. The methods used here are Laplace Transform method, method of separation of variables, Fourier Transform and MATLAB software. We reached the same solution at the end in Laplace Transform method, method of separation of variables, but by Fourier Transform we reached solution in different form that is in sine and cosine series form.

Thermal analysis of MHD convective slip transport of fractional Oldroyd-B fluid over a plate

The prime concern of this study is to analyze the generalized time-dependent magnetohydrodynamic (MHD) slip transport of an Oldroyd-B fluid near an oscillating upright plate. The plate is nested in a porous media under the action of ramped heating and nonlinear thermal radiation. Caputo–Fabrizio (CF) and Atangana–Baleanu (ABC) derivatives are utilized to constitute fractional partial differential equations that establish slip flow, shear stress, and heat transfer phenomena. Primarily, Laplace transformation is applied to dimensionless fractional models, and later Stehfest’s numerical algorithm is invoked to anticipate solutions of momentum and heat equations in principal coordinates. Moreover, computed solutions of velocity and energy fields are authenticated by Durbin’s and Zakian’s Laplace inversion algorithms. The relations for skin friction and Nusselt number are evaluated in terms of velocity and temperature gradients to efficiently anticipate shear stress and rate of heat transfer at the solid–fluid interface. The respective outcomes are manifested through tables. A critical examination of the current model is carried out and repercussions of variation in implanted parameters on temperature and momentum profiles are graphically elucidated. For the sake of comparison, three limiting fractional models, named second grade, Maxwell, and viscous models, are proposed for the isothermal and ramped temperature cases. Consequently, the observed outcomes affirm that under the isothermal condition, a generalized Maxwell fluid performs the swiftest slip transport compared to other models. Inversely, a second grade fluid specifies the highest velocity profile under ramped temperature case.

Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities

In this work, the local and global existence of weak solutions by using the Faedo-Galerkin method, the finite time blow up of the weak solution with positive initial energy and the general decay of the solution are discussed. Finally, we consider the exponential growth of the solution with sufficient conditions. This work generalizes and improves earlier results in the literature, see [L.X. Truong and N. Van Y, On a class of nonlinear heat equations with viscoelastic term, Comput. Math. Appl., 2016] and [L.X. Truong and N. Van Y, Exponential growth with ${L}^{p}$-norm of solutions for nonlinear heat equations with viscoelastic term, Appl. Math. Comput., 2016].

LOCAL APPROXIMATE SOLUTIONS OF A CLASS OF NONLINEAR DIFFUSION POPULATION MODELS

This paper studies approximate solutions for a class of nonlinear diffusion population models. Our methods are to use the fundamental solution of heat equations to construct integral forms of the models and the well-known Banach compression map theorem to prove the existence of positive solutions of integral equations. Non-steady-state local approximate solutions for suitable harvest functions are obtained by utilizing the approximation theorem of multivariate continuous functions.

Formula omitted] bounds of eigenfunctions with an indefinite weight function

This paper is concerned with the L∞ bounds of eigenfunctions to the eigenvalue problem Δu=−λm(x)uin Ω,u=0on ∂Ω,where Ω is a bounded domain and m(x) is an indefinite weight function. For the Dirichlet eigenvalue problem when m(x)≡1, the L∞ bound of eigenfunction is obtained by classical heat kernel estimate. However, the case is quite different for the weighted problem, additional difficulty appears if m(x) is indefinite. In this paper, we investigate the L∞ bounds of eigenfunctions by the means of comparison argument and fundamental solution of heat equation.

On Fractional Heat Equation

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.

Non-classical heat equation with singular memory term

In this paper, we consider the non-classical heat equation with singular memory term. This equation has many applications in various fields, for example liquids mechanics, solid mechanics, and heat conduction theory first, we prove that the solution exists locally in time. Then we investigate the converegence of the mild solution of non-classical heat equation, and the mild solution of classical heat equation.

Analytical Solution vs. Numerical Solution of Heat Equation Flow Through Rod of Length 8 Units in One Dimension

The paper presented the basic treatment of the solution of heat equation in one dimension. Heat is a form of energy in transaction and it flows from one system to another if there is a temperature difference between the systems. Heat flow is the main concern of sciences which seeks to predict the energy transfer which may take place between material bodies as result of temperature difference. Thus, there are three modes of heat transfer, i.e., conduction, radiation and convection. Conduction can be steady state heat conduction, or unsteady state heat conduction. If the system is in steady state, temperature doesn’t vary with time, but if the system in unsteady state temperature may varies with time. However, if the temperature of material is changing with time or if there are heat sources or sinks within the material the situation is more complex. So, rather than to escape all problem, we are targeted to solve one problem of heat equation in one dimension. The treatment was from both the analytical and the numerical view point, so that the reader is afforded the insight that is gained from analytical solution as well as the numerical solution that must often be used in practice. Analytical we used the techniques of separation of variables. It is worthwhile to mention here that, analytical solution is not always possible to obtain; indeed, in many instants they are very cumber some and difficult to use. In that case a numerical technique is more appropriate. Among numerical techniques finite difference schema is used. In both approach we found a solution which agrees up to one decimal place.

Numerical method for pricing discretely monitored double barrier option by orthogonal projection method

<abstract> In this paper, we consider discretely monitored double barrier option based on the Black-Scholes partial differential equation. In this scenario, the option price can be computed recursively upon the heat equation solution. Thus we propose a numerical solution by projection method. We implement this method by considering the Chebyshev polynomials of the second kind. Finally, numerical examples are carried out to show accuracy of the presented method and demonstrate acceptable accordance of our method with other benchmark methods. </abstract>

Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity

<p style='text-indent:20px;'>We are concerned with blow-up mechanisms in a semilinear heat equation: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t = \Delta u + |x|^{2a} u^p , \quad x \in \textbf{R}^N , \, t&gt;0, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ a&gt;-1 $\end{document}</tex-math></inline-formula> are constants. As for the Fujita equation, which corresponds to <inline-formula><tex-math id="M3">\begin{document}$ a = 0 $\end{document}</tex-math></inline-formula>, a well-known result due to M. A. Herrero and J. J. L. Velázquez, C. R. Acad. Sci. Paris Sér. I Math. (1994), states that if <inline-formula><tex-math id="M4">\begin{document}$ N\geq 11 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ p&gt; 1 + 4/(N-4-2\sqrt{N-1}) $\end{document}</tex-math></inline-formula>, then there exist radial blow-up solutions <inline-formula><tex-math id="M6">\begin{document}$ u_{\ell, {\rm HV}}(x, t) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \ell \in \bf{N} $\end{document}</tex-math></inline-formula>, such that <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \lim\limits_{t\to T} \left( T-t \right)^{1/(p-1)} \| u_{\ell, {\rm HV}}(\cdot, t) \|_{L^{\infty}(\textbf{R}^N )} = \infty, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M8">\begin{document}$ T $\end{document}</tex-math></inline-formula> is the blow-up time. We revisit the idea of their construction and obtain refined estimates for such solutions by the techniques developed in recent works and elaborate estimates of the heat semigroup in backward similarity variables. Our method is naturally extended to the case <inline-formula><tex-math id="M9">\begin{document}$ a\not = 0 $\end{document}</tex-math></inline-formula>. As a consequence, we obtain an example of solutions that blow up at <inline-formula><tex-math id="M10">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>, the zero point of potential <inline-formula><tex-math id="M11">\begin{document}$ |x|^{2a} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M12">\begin{document}$ a&gt;0 $\end{document}</tex-math></inline-formula>, and behave in non-self-similar manner for <inline-formula><tex-math id="M13">\begin{document}$ N &gt; 10 + 8a $\end{document}</tex-math></inline-formula>. This last result is in contrast to backward self-similar solutions previously obtained for <inline-formula><tex-math id="M14">\begin{document}$ N &lt; 10 + 8a $\end{document}</tex-math></inline-formula>, which blow up at <inline-formula><tex-math id="M15">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>.

One dimensional thermal shock problem in a thick plate

In this research, investigation carried out for thermo-elastic response of thick solid circular plate subjected to certain boundaries. According to the linear function of temperature, heat source are generated further. For finding the solution of heat equation, the integral transform techniques are used. Hankel transform techniques are incorporated for analysis of the numerical results. For designing of engineering applications of beneficial structures or machines, the temperature, displacement and thermal stresses are investigated. Suitable values are provided to investigate the parameters and the expression for specific case of special interest and the numerical results has been investigated.

Theoretical and experimental investigation of spatial temperature distribution in active fiber under conditions of laser radiation generation

The longitudinal and transversal temperature distribution in active waveguide, doped with Yb/Er ions, under conditions of laser generation was calculated as the composition of two mathematical models: computation of longitudinal pump radiation distribution by solution of rate equations for Yb/Er doped active medium and computation of transversal temperature distribution by solution of heat equation for convectively cooled fiber using finite element method. The theoretical results were in a good agreement with experimental data, measured by the piezoelectric resonators used as temperature sensors, placed on the surface of the fiber. The pump radiation transferring length in active waveguide and the location of the most heated fiber area, as well as the maximum temperature value, were calculated using a created model of pump radiation transferring into GT-wave active fiber.