Numerical Solution of Parabolic Partial Differential Equation by Using Finite Element Method

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.

The scope of this book is to present well known simple and advanced numerical methods for solving partial differential equations (PDEs) and how to implement these methods using the programming environment of the software package Diffpack. A basic background in PDEs and numerical methods is required by the potential reader. Further, a basic knowledge of the finite element method and its implementation in one and two space dimensions is required. The authors claim that no prior knowledge of the package Diffpack is required, which is true, but the reader should be at least familiar with an object oriented programming language like C++ in order to better comprehend the programming environment of Diffpack. Certainly, a prior knowledge or usage of Diffpack would be a great advantage to the reader.

This study explores the analytical and numerical solutions of partial differential equations (PDEs), focusing on parabolic (heat). The first part presents their analytical solutions using initial and boundary conditions and delves into the finite difference method (FDM), discussing forward, backward, and central difference schemes. These methods are applied to numerically solve one- and two-dimensional heat. The Crank-Nicolson method, recognized for its unconditional stability, is employed to improve the accuracy of heat equation solutions, overcoming limitations of explicit and implicit schemes. We then analyze the performance, strengths, and weaknesses of FDM through numerical simulations of one-dimensional heat. Due to computational constraints, Crank-Nicolson for 1D simulation, was not executed. Results indicate that the implicit backward difference method demonstrates superior stability by allowing unrestricted step sizes compared to the explicit forward difference method. These findings contribute to a deeper understanding of numerical PDE solutions and stability considerations in computational mathematics.

A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The analytical solution is needed to obtain the exact solution of partial differential equation. To solve these partial differential equations, the appropriate boundary and initial conditions are needed. The general solution is dependent not only on the equation, but also on the boundary conditions. In other words, these partial differential equations will have different general solution when paired with different sets of boundary conditions. In the present study, the homogeneous one-dimensional heat equation will be solved analytically by using separation of variables method. Our main objective is to determine the general and specific solution of heat equation based on analytical solution. To verify our objective, the heat equation will be solved based on the different functions of initial conditions on Neumann boundary conditions. The results have been compared with different values of initial conditions but the boundary condition remain the same. Based on the results obtained, it can be concluded that increase the number of n will reduce the heat temperature and the time taken. For short length of the rod, the heat temperature quickly converges to zero and take less time to release or reduced the heat temperature when compared to the long length of the rod.

The objective of this paper is to examine the utility of direct, numerical solution procedures, such as finite difference or finite element methods, for partial differential equations in chaotic dynamics. In the past, procedures for solving such equations to detect chaos have utilized Galerkin approximations which reduce the partial differential equations to a set of truncated, nonlinear ordinary differential equations. This paper will demonstrate that a finite difference solution is equivalent to a Galerkin solution, and that the finite difference method is more powerful in that it may be applied to problems for which the Galerkin approximations would be difficult, if not impossible to use. In particular, a nonlinear partial differential equation which models a slender, Euler-Bernoulli beam in compression issolvedto investigate chaotic motions under periodic transverse forcing. The equation, cast as a system of firstorder partial differential equations is directly solved by an explicit finite difference scheme. The numerical solutions are shown to be the same as the solutions of an ordinary differential equation approximating the first mode vibration of the buckled beam. Then rigid stops of finite length are incorporated into the model to demonstrate a problem in which the Galerkin procedure is not applicable. The finite difference method, however, can be used to study the stop problem with prescribed restrictions over a selected subdomain of the beam. Results obtained are briefly discussed. The end result is a more general solution technique applicable to problems in chaotic dynamics.

In the present and following chapters extensive use will be made of a simple finite element code mlfem_nac. This code, including a manual, can be freely downloaded from the website: www.mate.tue.nl/biomechanicsbook. The code is written in the program environment MATLAB. To be able to use this environment a licence for MATLAB has to be obtained. For information about MATLAB see: www.mathworks.com. Introduction It will be clear from the previous chapters that many problems in biomechanics are described by (sets of) partial differential equations and for most problems it is difficult or impossible to derive closed form (analytical) solutions. However, by means of computers, approximate solutions can be determined for a very large range of complex problems, which is one of the reasons why biomechanics as a discipline has grown so fast in the last three decades. These computer-aided solutions are called numerical solutions, as opposed to analytical or closed form solutions of equations. The present and following chapters are devoted to the numerical solution of partial differential equations, for which several methods exist. The most important ones are the Finite Difference Method and the Finite Element Method. The latter is especially suitable for partial differential equations on domains with complicated geometries, material properties and boundary conditions (which is nearly always the case in biomechanics). That is why the next chapters focus on the Finite Element Method. The basic concepts of the method are explained in the present chapter.

This is another book on the finite element method for the approximation of partial differential equations. Its goal is to get the reader acquainted with the basic finite element techniques that allow him (or her) to program the method. The presentation is coherently carried out to achieve this goal. In Part I a simple model boundary value problem, the Laplace equation with Dirichlet conditions, is used as paradigm to illustrate the mathematical aspects of the method. Then the data structure for mesh generation is introduced, the stiffness matrix and the right hand side assembled, and several numerical algorithms are proposed and programmed for the solution of the linear system by either direct and iterative methods. The programming languages are Fortran 77 and C. This part is very elementary, and therefore very useful to beginners in finite elements. In Part II the previous arguments are extended to the case of more general elliptic equations of second order, as well as to linear evolution equations such as the heat equation and the wave equation. Here the finite difference schemes are used for time-discretization, combined with finite element (or finite volume) for space-discretization. Part III addresses more heterogeneous issues, remarkably the boundary integral method for the Laplace equation, and some domain decomposition methods for parallel computing. The mortar method for treating non-conforming grid partitioning is also sketched. As stated in the title, this has to be considered an introductory volume to the subject of computing the solution of simple partial differential equations by the finite element method. Essentially no analysis is carried out for complex (nonlinear) problems. However, the clarity of exposition, and the continuous interplay between mathematical equations, finite element programming and illustrative examples, make this book a very valuable tool, especially for students in applied mathematics and engineering.

Partial differential equations (PDEs) are fundamental in describing various physical phenomena, such as fluid dynamics, heat conduction, and wave propagation. However, analytical solutions to these equations are often difficult or impossible to obtain due to their complexity and the boundary conditions involved. Numerical methods provide an effective alternative by approximating solutions through discretization techniques. This paper explores various numerical methods for solving PDEs, including finite difference, finite element, and finite volume methods. We discuss their theoretical foundations, implementation strategies, and advantages in handling different types of PDEs, such as elliptic, parabolic, and hyperbolic equations. Moreover, the paper addresses key challenges such as stability, convergence, and computational efficiency, and reviews the use of high-performance computing in tackling large-scale problems. The applications of these methods in scientific computing and engineering are highlighted, demonstrating their versatility and importance in solving real-world problems.The numerical solution of partial differential equations (PDEs) plays a crucial role in solving real-world problems across various fields, including physics, engineering, and finance. Exact analytical solutions to PDEs are often not feasible due to their complexity and the nature of boundary conditions. As a result, numerical methods such as the finite difference, finite element, and finite volume methods are widely employed to approximate solutions. This paper provides an overview of these methods, emphasizing their formulation, implementation, and application to different types of PDEs, including elliptic, parabolic, and hyperbolic equations. Key considerations such as stability, convergence, and accuracy are discussed, along with strategies for improving computational efficiency. The paper also highlights the use of advanced computational techniques and parallel computing in addressing large-scale and complex PDE systems. Overall, numerical methods offer powerful tools for solving PDEs and are essential for simulating and analyzing complex phenomena in science and engineering.

For a very long time, finite volume, finite element, or finite difference methods have been used to solve partial differential equations (PDEs) numerically. These techniques have been used by researchers for centuries to solve a wide range of mathematical, physical, or chemical problems. The complexity of these numerical approaches, for the resolution of the PDEs in space dimensions equal to two or higher, can come from the coding, the management, and the good choice of the triangulation or the mesh of the domain in which one wishes to locate the solution. The radial basis function collocation method is a meshless technique used to numerically solve some partial differential equations and is based on the nodes of the domain and a radial basis function is a real-valued function whose value only depends on the separation of its input parameter x from another fixed point, sometimes known as the function's origin or center. This method was introduced by KANSA in the 1990s. In this study, the numerical simulation of the one-dimensional heat equation was carried out using the RBF Collocation Method and particularly the Gaussian function. This model was used to test the accuracy and efficiency of this method by comparing numerical and analytical solutions on rectangular geometry with collocation nodes. The results show that the RBF collocation approximate solution and the exact solution coincided in test case problems 2, 3 and 4.

Whereas the previous chapters are exclusively dedicated to lumped systems (systems of dimension 0 described by ODEs) and distributed parameter systems in one spatial dimension, this chapter touches upon the important class of problems in more space dimensions, as well as problems with time-varying spatial domains. Both are difficult topics and the ambition of this chapter is just to give a foretaste of possible numerical approaches. Finite difference schemes on simple 2D domains, such as squares, rectangles or more generally convex quadrilaterals, are first introduced, including several examples such as the heat equation, Graetz problem, a tubular chemical reactor, and Burgers equation. Finite element methods, which have more potential than finite difference schemes when considering problems in 2D, are then discussed based on a particular example, namely FitzHugh-Nagumo model. This example also gives the opportunity to apply the proper orthogonal decomposition method to derive reduced-order models. Finally, the problematic of time-varying domains is introduced via another particular application example related to freeze drying. The main idea here is to use a transformation so as to convert the original problem into a conventional one with a time-invariant domain.Whereas the previous chapters are exclusively dedicated to lumped systems (systems of dimension 0 described by ODEs) and distributed parameter systems in one spatial dimension, this chapter touches upon the important class of problems in more space dimensions, as well as problems with time-varying spatial domains. Both are difficult topics and the ambition of this chapter is just to give a foretaste of possible numerical approaches. Finite difference schemes on simple 2D domains, such as squares, rectangles or more generally convex quadrilaterals, are first introduced, including several examples such as the heat equation, Graetz problem, a tubular chemical reactor, and Burgers equation. Finite element methods, which have more potential than finite difference schemes when considering problems in 2D, are then discussed based on a particular example, namely FitzHugh-Nagumo model. This example also gives the opportunity to apply the proper orthogonal decomposition method to derive reduced-order models. Finally, the problematic of time-varying domains is introduced via another particular application example related to freeze drying. The main idea here is to use a transformation so as to convert the original problem into a conventional one with a time-invariant domain.

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The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

Adaptive Solution of Partial Differential Equations in Multiwavelet Bases

This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.

Рассматривается подход к моделированию динамики распространения тепла по объему гомогенного материала внутри реакционной камеры пиролизного регенератора при наличии нагревателей, размещаемых внутри камеры. Проанализированы несколько подходов для расчета температурных полей: аналитический (непосредственное решение дифференциального уравнения Фурье в частных производных), численный (метод конечных разностей и метод конечных элементов), применение эквивалентных электрических схем и компьютерное моделирование. Показано, что данная динамика описывается дифференциальными уравнениями дробного порядка. При этом форма уравнений, описывающих зависимости токов в ветвях электрической цепи и напряжения в ее узлах, аналогична форме уравнений, описывающих зависимость теплового потока в среде и значений температуры в отдельных ее точках. Таким образом, решение дифференциального уравнения заменяется на моделирование работы электрической цепи во временнóй области. Предложены схемотехнические модели теплопроводности среды для таких элементов пространства, как стержень, а на его основе - элемент плоскости, столбец и объем. С помощью данных элементов проведено моделирование нестационарного распространения температуры по объему среды при наличии от одного до трех нагревательных элементов внутри объема. Корректность схемотехнического моделирования подтверждена с помощью специализированного ПО, реализующего классический метод конечных элементов.