Applicability extent of 2-D heat equation for numerical analysis of a multiphysics problem

Some strategies for solving differential equations based on the finite difference method are presented: forward time centered space (FTSC), backward time centered space (BTSC), and the Crank-Nicolson scheme (CN). These are developed and applied to a simple problem involving the one-dimensional (1D) (one spatial and one temporal dimension) heat equation in a thin bar. The numerical implementation in this work can be used as a preamble to introduce a method of solving the heat equation that can be implemented in problems in the area of finances. The results of implementing the software on very fine meshes (unidimensional), and with relatively small-time steps, are shown. Through mesh refinement, it was possible to obtain a better temperature distribution in the thin bar between a range of points. The heat equation was solved numerically by testing both implicit (CN) and explicit (FTSC and BTSC) methods. The examples show that the implemented schemes conform to theoretical predictions and that truncation errors depend on mesh, spacing, and time step.

Composite materials are becoming more popular in technological applications due to the significant weight savings and strength offered by these materials compared to metallic materials. In many of these practical situations, the structures suffer from drop-impact loads. Materials and structures significantly change their behavior when submitted to impact loading conditions compared to quasi-static loading. The present work is devoted to investigating the thermal process in carbon-fiber-reinforced polymers (CFRP) subjected to a drop test. A novel drop-weight impact test experiment is performed to evaluate parameters specific to 3D composite materials. A strain gauge rosette and infrared thermography are employed to record the kinematic and thermal fields on the composites’ surfaces. This technique is nondestructive and offers an extensive full-field investigation of a material’s response. The combination of strain and infrared thermography data allows a comprehensive analysis of thermoelastic effects in CFRP when subjected to impacts. The experimental results are validated using numerical analysis by developing a MATLAB® code to analyze whether the coupled heat and wave equation phenomenon exists in a two-dimensional polar coordinate system by discretizing through a forward-time central-space (FTCS) finite-difference method (FDM). The results show the coupling has no significant impact as the waves generated due to impact disappears in 0.015 s. In contrast, heat diffusion happens for over a one-second period. This study demonstrates that the heat equation alone governs the CFRP heat flow process, and the thermoelastic effect is negligible for the specific drop-weight impact load.

Two finite difference schemes, FTBSCS (forward time backward space and centered space) and FTCS (forward time centered space) have been studied for solving convection-diffusion equation (CDE) with appropriate initial condition and boundary conditions. The convection velocity u(t, x) of CDE is computed by solving viscous Burger’s equation using the same schemes. Stability conditions of the schemes are determined and it is analytically shown that FTCS scheme is superior to FTBSCS scheme in terms of time step selection. The conditions of stability are also numerically verified. Some numerical simulation results are presented for various parameters. Error comparisons of both the schemes are presented to estimate accuracy of solutions.

Recent developments in complex fuzzy (CF) sets have extended the classical fuzzy framework from the unit interval [0,1] to the unit disk in the complex plane C, allowing for the modeling of uncertainties in both magnitude and phase. Building upon this foundation, this study introduces-for the first time-the use of complex intuitionistic fuzzy (CIF) numbers to solve partial differential equations, specifically focusing on the CIF heat equation. The CIF framework integrates both membership and non-membership functions with a complex-valued representation, enabling a more expressive treatment of uncertainty, including hesitation. An explicit finite difference method, namely the Forward Time Central Space (FTCS) scheme, is employed to discretize and solve the CIF heat equation. The model considers fuzziness in the initial and boundary conditions, where uncertainty impacts amplitude and phase terms. To represent this uncertainty, triangular fuzzy numbers are used for the real and imaginary parts within the complex unit disk. The proposed approach demonstrates numerical stability and achieves second-order spatial and first-order temporal accuracy, validating its reliability and effectiveness. A numerical example confirms the feasibility of the method, showing strong alignment with theoretical predictions. This work generalizes existing CF heat equation models and provides a foundation for solving more complex systems involving higher-order and bipolar uncertainties in future studies.

All materials have different and unique thermal properties that determine how the temperature changes when a material is subjected to a temperature difference. This study was intended to investigate the thermal properties of a polymer called Polyurethane, focusing on anti-seepage and anti-abrasion polyurethane. The thermal conductivity and heat transfer coefficient of cold polyurethane specimens have been calculated by capturing the infrared signature using a FLIR T1030sc Infrared camera and comparing the results with simulated results. The simulations were carried out in MATLAB®, and the solution is based on the Heat equation. This paper describes the driving mechanisms behind the Heat equation and how the approximated solution to the Heat equation is obtained by discretizing through a forward-time central-space (FTCS) finite-difference method. The results reveal that the heat transfer coefficient for anti-abrasion Polyurethane is almost four times that for anti-seepage Polyurethane. The thermal conductivity for the respective has a difference of a factor of two. A good agreement between the experimental and the numerical study was acheived. This study is helpful for the potential use of polyurethane material in Arctic regions either as a coating material for pipes or as a sealent in the oil and gas industry.

One-dimensional heat equation was solved for different higher-order finite difference schemes, namely, forward time and fourth-order centered space explicit method, backward time and fourth-order centered space implicit method, and fourth-order implicit Crank-Nicolson finite difference method. Higher-order schemes have complexity in computing values at the neighboring points to the boundaries. It is required there a specification of the values of field variables at some points exterior to the domain. The complexity was incorporated using Hicks approximation. The convergence and stability analysis was also computed for those higher-order finite difference explicit and implicit methods in case of solving a one dimensional heat equation. The obtained numerical results were compared with exact solutions. It is found that backward time and fourth-order centered space implicit scheme along with Hicks approximation performed well over the other mentioned higher-order approaches.

Partial differential equations (PDEs) are used in the real world to model physical phe- nomena such as heat, wave, Laplace, and Poisson equations. For regular shape domains, the heat equation can be solved analytically; however, for irregular domains, the computation of the solu- tion is difficult and numerical methods like Finite Difference Method (FDM) and Finite Element Method (FEM) can be used. FEM provides approximate values at discrete points in the domain. It breaks down a large problem into smaller finite elements. These element’s equations are combined into a system representing the whole problem. We show the comparison between analytic solution, solutions by FDM and FEM. The impact of heat on the material is examined at various positions and multiple positions. We compare the analytical and numerical (by FEM) solution considering several homogeneous materials with various diffusivity values (α). Finally, the simulation results of different non-homogeneous materials were compared. Science and engineering fields that use heat equations can be evaluated using the numerical method applied here.

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.

Solutions of heat or diffusion equations with the boundary conditions which is a dynamic random field are discussed. This kind of method can be used to obtain the description of heat equations or diffusion equations based on observed physical reality, ie ordinary differential equations, representing heat or diffusion propagation, with a boundary condition that satisfies stochastic differential equations. The heat or diffusion equations obtained from the method are the compared to the heat equation or the stochastic diffusion. The comparison is emphasized on the existence and properties of Green functions.

In this paper, we obtain the numerical solution of a 1-D generalised Burgers-Huxley equation under specified initial and boundary conditions, considered in three different regimes. The methods are Forward Time Central Space (FTCS) and a non-standard finite difference scheme (NSFD). We showed the schemes satisfy the generic requirements of the finite difference method in solving a particular problem. There are two proposed solutions for this problem and we show that one of the proposed solutions contains a minor error. We present results using FTCS, NSFD, and exact solution as well as show how the profiles differ when the two proposed solutions are used. In this problem, the boundary conditions are obtained from the proposed solutions. Error analysis and convergence tests are performed.

An inverse problem concerning the two-dimensional diffusion equation with source control parameter is considered. Four finite-difference schemes are presented for identifying the con- trol parameter which produces, at any given time, a desired energy distribution in a portion of the spatial domain. The fully explicit schemes developed for this purpose, are based on the (1,5) forward time centred space (FTCS) explicit formula, and the (1,9) FTCS scheme, are economical to use, are second-order and have bounded range of stability. Therange of stability for the 9-point finite difference scheme is less restrictive than the (1,5) FTCS formula. The fully implicit finite difference schemes employed, are based on the (5,1) backward time centred space (BTCS) formula, and the (5,5) Crank–Nicolson implicit scheme, which are unconditionally stable, but use more CPU times than the fully explicit techniques. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments are presented, and central processor (CPU) times needed for solving this inverse problem are reported.

Option can be defined as a contract between two sides/parties said party one and party two. Party one has the right to buy or sell of stock to party two. Party two can invest by observe the put option price or call option price on a time period in the option contract. Black-Scholes option solution using finite difference method based on forward time central space (FTCS) can be used as the reference for party two in the investment determining. Option price determining by using Black-Scholes was applied on Samsung stock (SSNLF) by using finite difference method FTCS. Daily data of Samsung stock in one year was processed to obtain the volatility of the stock. Then, the call option and put option are calculated by using FTCS method after discretization on the Black-Scholes model. The value of call option was obtained as $1.457695030014260 and the put option value was obtained as $1.476925604670225.

Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains

Fourth-order techniques for identifying a control parameter in the parabolic equations

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.