Two-dimensional Heat Equation Research Articles

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach.

The Two-Dimensional Conduction Heat Transfer Equation on a Square Plate: Explicit vs. Crank-Nicolson Method in MS Excel Spreadsheet

In this paper, the two-dimensional conduction heat transfer equation on a square plate is analyzed using a finite difference method. We have developed both the forward time-centered space (FTCS) and Crank-Nicolson (CN) finite difference schemes for the two-dimensional heat equation, employing Taylor series. Subsequently, these schemes were employed to solve the governing equations. The primary objective of this study is to compare the efficiency of the two methods in solving the conduction heat transfer equation. This was accomplished by implementing Spreadsheet Excel instructions. The results are presented, highlighting a comparison between the exact and approximate solutions. Furthermore, to demonstrate the convergence of the numerical schemes, we estimated the error between the actual and approximate solutions for a specific numerical problem and presented the results graphically. The data utilized in this research included the thermal conductivity of the medium of the square plate concerning width, grid, and compliance with initial and boundary conditions. The findings indicate that the Crank-Nicolson method is more accurate than the forward time-centered space method, as it approaches the exact solution more effectively. Furthermore, this study confirms that the solution and simulation of the heat transfer equation on a square plate can be accurately performed using an Excel spreadsheet as well as other numerical software.

Solution to a Two-Dimensional Nonlinear Parabolic Heat Equation Subject to a Boundary Condition Specified on a Moving Manifold

Solution to a Two-Dimensional Nonlinear Parabolic Heat Equation Subject to a Boundary Condition Specified on a Moving Manifold

Delta-Bose gas from the viewpoint of the two-dimensional stochastic heat equation

Delta-Bose gas from the viewpoint of the two-dimensional stochastic heat equation

Analytical solution of fuzzy heat problem in two-dimensional case under Caputo-type fractional derivative.

This work aims to investigate the analytical solution of a two-dimensional fuzzy fractional-ordered heat equation that includes an external diffusion source factor. We develop the Sawi homotopy perturbation transform scheme (SHPTS) by merging the Sawi transform and the homotopy perturbation scheme. The fractional derivatives are examined in Caputo sense. The novelty and innovation of this study originate from the fact that this technique has never been tested for two-dimensional fuzzy fractional ordered heat problems. We presented two distinguished examples to validate our scheme, and the solutions are in fuzzy form. We also exhibit contour and surface plots for the lower and upper bound solutions of two-dimensional fuzzy fractional-ordered heat problems. The results show that this approach works quite well for resolving fuzzy fractional situations.

Unsteady State Temperature Distribution Inside House Based on Slope Roof

The house is a building that serves as a place to gather with family and also to get comfort. The house is designed so the occupants can feel aesthetically and functionally comfortable. It will be interesting to discuss the temperature inside the house as one of the comfort factors. The study aims to analyze the temperature conditions inside the house, which are influenced by the slope of the roof. A two-dimensional unsteady state heat equation represents this temperature since the temperature distribution satisfies the heat transfer concept and changes over time. This governing equation will be approximated by the forward time center space scheme as one finite difference method and completed by the quadratic ghost point method since the house domain is an irregular shape. Van Neumann criteria are applied here to analyze the stability of the computational approach for this numerical scheme. Furthermore, these schemes are implemented in MATLAB application to quantitatively and visually display temperature dynamics. Some simulations completed these approximations to see temperature variations over time. The results show that the bigger slope causes the average temperature to be cooler. In other words, the temperature in the house will be more comfortable when the roof slope gets bigger.

Direct and inverse problems for a 2D heat equation with a Dirichlet–Neumann–Wentzell boundary condition

In this paper we present the inverse problem of determining a time dependent heat source in a two-dimensional heat equation accompanied with Dirichlet–Neumann–Wentzell boundary conditions. The model is of significant practical importance in applications where the time dependent internal source is to be controlled from total energy measurements in the case when the boundary is defined with partially absorbing (Dirichlet), partially reflecting (Neumann), and partially having the capacity for storing heat (Wentzell) conditions. The considered spatial domain is a rectangle which allows the application of a Fourier series analysis to the solution of the problem whose auxiliary spectral problem is a novel eigenproblem. This eigenvalue problem is defined using the two-dimensional Laplace operator where the spectral parameter resides in one of the accompanying boundary conditions. Under this setting we study the well-posedness of both the direct and inverse problems. In addition, we use the results obtained on the well-posedness to establish the numerical solvability of all the problems considered in the paper. In order to achieve this, we make use of a Chebyshev spectral collocation method for the spatial discretisation and a backward differentiation formula for the temporal discretisation. All the numerical results are shown to be consistent with the theory and in good agreement with the analytical solutions.

Study on Sublimation Drying of Carrot and Simulation by Using Cellular Automata

Vacuum freeze-dried products exhibit properties characteristic of porous media, rendering them superior in both drying and rehydration capabilities. However, the process of sublimation drying is constrained by its substantial time and energy costs. To comprehensively grasp its technological process and identify the optimal process parameters, the cellular automata method was employed for sublimation process simulation. Carrot slices, measuring 10 mm in thickness and 40 mm in radius, were selected for both simulation and experimentation. The sublimation process was characterized using a two-dimensional heat and mass transfer equation, inclusive of a dusty gas model. Additionally, a cellular automaton model was applied to simulate the mass transfer process, temperature, and moisture content changes in the sublimation drying stage. Then, the accuracy of the model was verified through experimentation. There was a remarkable alignment between simulation and experimental outcomes, with determination coefficients R2 of 99.4% for moisture content and 97.6% for temperature variations.

Probabilistic analysis of a class of 2D-random heat equations via densities

We give new probabilistic results for a class of random two-dimensional homogeneous heat equations with mixed homogeneous Dirichlet and Neumann boundary conditions and an arbitrary initial condition on a rectangular domain. The diffusion coefficient is assumed to be an arbitrary second-order random variable, while the initial condition is a stochastic process admitting a Karhunen-Loève expansion. We then construct pointwise convergent approximations for the main moments and the density of the solution. The theoretical results are numerically illustrated.

Physical restriction neural networks with restarting strategy for solving mathematical model of thermal heat equation for early diagnose breast cancer

Breast cancer is a serious health concern for women in both developed and developing countries, and early diagnosis is crucial for an effective treatment. One possible approach of detection is by observing abnormal surface temperatures of the breast. To study the thermal behavior of the human breast, this research paper employs the two-dimensional heat equation’s partial differential equation (PDE). In fact, we propose a method called r-PINN-Adam, which uses Physics-Informed Neural Networks (PINN) with a restarting strategy to solve the PDE thermal analysis. PINN is a method that incorporates physical law in each dataset to solve PDE problems, while Adam is used to update the weights of the ANN and minimize the loss function. The restarting process monitors the progress of the loss values, and if no improvement is made, the process is restarted with the best weights as the initial input to the next cycle. This approach ensures that the method finds the smallest value of the loss function, and it is also more efficient as no time is wasted on iterations that do not significantly improve the results. The PDE thermal analysis was solved for normal and tumorous breasts to study the temperature behavior surrounding the cancer. The proposed r-PINN-Adam method was compared to the basic PINN and other optimizers in terms of accuracy and efficiency. The numerical results indicated that our proposed method yields competitive results compared to state-of-the-art methods, while significantly reducing computational time.

The Geometry of Vector Fields and Two Dimensional Heat Equation

The geometry of orbits of families of smooth vector fields was studied by many mathematicians due to its importance in applications in the theory of control systems, in dynamic systems, in geometry and in the theory of foliations. In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate singular foliation every regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields.

Lie algebra and conservation laws for the time-fractional heat equation

The Lie symmetry method is applied to derive the point symmetries for the N-dimensional fractional heat equation. We find that the numbers of symmetries and Lie brackets are reduced significantly as compared to the integer order for all dimensions. In fact for integer order linear heat equation the number of solution symmetries is equal to the product of the order and space dimension, whereas for the fractional case, it is half of the product on the order and space dimension. The Lie algebras for the integer and fractional order equations are mentioned using the subsequent computations of Lie brackets and by inspection. Interestingly, it is observed that for the one-dimensional fractional heat equation, the Lie algebra obtained by inspection of symmetries is similar to the result obtained by computation of Lie brackets, which is . The Lie algebra using the symmetries of the two-dimensional heat equation is observed to be , whereas using the Lie brackets the algebra is deduced to be . Hence, it can be concluded that the Lie algebra obtained from the nonzero Lie brackets can be conflated to the algebra which is obtained by inspection. Further, the subsequent Lie algebras are mentioned for the three and four-dimensional integer and fractional equations and the conservation laws are explicitly stated.

Design and performance analysis of salinity gradient solar pond under different climatic and soil conditions.

A salinity gradient solar pond (SGSP) is capable of storing a significant quantity of heat for an extended period of time. It is a great option for providing hot water at a reduced energy cost. Additionally, SGSP is used in low-temperature industrial applications such as saltwater desalination, space heating, and power generation. Solar pond thermal performance is dependent on a variety of operational variables, including the soil conditions, the climate of the particular site, the thickness of the solar pond layers, the depth of the water table, and the salt content of the pond. As such, this study examines the thermal performance of a solar pond under a variety of operational conditions. The solar pond model is used to test the thermal performance by simulating two-dimensional heat and mass transport equations. The equations are solved using the finite difference technique utilizing MATLAB® scripts. Salt distributions and temperature profiles are computed for a variety of factors influencing SGSP's thermal performance. The main distinguishing variables influencing the thermal performance of SGSP are soil conditions, such as soil texture, types, the moisture level in soil, and water table depth. The final findings indicated that the fine sand dry soil performed better than the other soil types owing to its poor heat conductivity. The economic results indicated that the period of return (POR) of the intended system is around 2 years. The solar pond construction costs such as excavation, transportation, salt and lining, were considered based on the local prices. This modeled study extracted the greatest possible energy is 110W/m2, with the fine sand dry at 62.48°C lowest temperature. This study suggested that the climatic conditions of Lahore is better than climatic conditions of Islamabad. Additionally, deeper water tables are suggested for improved thermal performance of the pond.

Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions

The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transformation h=logu, where u is the solution of the multiplicative stochastic heat equation(SHE). In [CD20, CSZ20, G20], they consider the solution of two dimensional KPZ equation via the solution uε of SHE with the flat initial condition and with noise which is mollified in space on scale in ε and its strength is weakened as βε=βˆ 2π −logε, and they prove that when βˆ∈(0,1), 1 βε(loguε−E[loguε]) converges in distribution as a random field to a solution of Edwards-Wilkinson equation. In this paper, we consider a stochastic heat equation uε with a general initial condition u0 and its transformation F(uε) for F in a class of functions F, which contains F(x)=xp (0<p≤1) and F(x)=logx. Then, we prove that 1 βε(F(uε(t,⋅))−E[F(uε(t,⋅))]) converges in distribution as a random field to a centered Gaussian field jointly in finitely many F∈F, t, and u0. In particular, we show the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depend on u0. Our main tools are Itô’s formula, the martingale central limit theorem, and the homogenization argument as in [CNN22]. To this end, we also prove a local limit theorem for the partition function of intermediate disorder 2d directed polymers.

Experimental and analytical method for determining specific losses in the presence of a surface effect

In this article, based on the accumulated experience of determining the intensity of the internal loss function in the active nodes of electromechanical devices, an experimental and analytical method for determining this intensity in time by the coordinates of the area under study is proposed. The error in determining the intensity of internal losses is determined based on comparing the results of calculating the integration of two variants of the two-dimensional heat equation.

Temperature stresses in a rectangular two-layer plate under the action of a locally distributed temperature field

A rectangular isotropic two-layer plate of an irregular structure is considered, the edges of which are freely supported, and a constant temperature is maintained on them. Two-dimensional Kirchhoff-type thermoelasticity equations and two-dimensional heat equations written for an inhomogeneous material were used to study the temperature stresses in the plate. Using the method of double trigonometric series in spatial variables and the Laplace integral transformation over time, the general solutions of boundary value problems of thermoelasticity and heat conductivity for this plate under the action of a locally distributed temperature field specified at the initial moment of time are written down. The normal stresses in the layers of the plate are numerically analyzed depending on the geometric parameters, heat transfer coefficient, and time.

Vector fields and differential equations

The geometry of orbits of families of smooth vector fields is an important object of mathematics due to its importance in applications, in the theory of dynamic systems and in the foliation theory. The paper devoted to the applications of the geometry of orbits of vector fields in four dimensional Euclidean space in theory of differential equations. It is shown that orbits generate singular foliation ever regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields. In this paper smoothness is smoothness of the class C ∞.

Analysis of Two-Dimensional Heat Transfer Problem Using the Boundary Integral Equation

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.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

STUDY OF THE THERMAL RESISTANCE OF HEAT TRANSFER IN TRANSIENT DYNAMIC REGIME IN AN INSULATING MATERIAL BASED ON TYPHA AND TWO-DIMENSIONAL (2D) CLAY: INFLUENCE OF THE HEAT EXCHANGE COEFFICIENT ON THE FRONT FACE

The objective of this study is to focus on the influence of local insulating materials in thermal insulation for a well-defined period. The material is a wall consisting of typha and clay. The main objective of this study is to see the behavior of the thermal resistance of the material as a function of the heat exchange coefficient at the front face. The typha and clay wall is modeled by applying the transient analytical method with the two-dimensional heat equation. This analytical resolution of the heat equation allows us to determine the detailed expression of the thermal resistance but also to visualize the curves of evolution of this resistance according to the depth under the influence of the heat exchange coefficient at the front side.