Tag: partial differential equation
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Heat Transfer Analysis and Open Sourced Finite Element Tools (2) About Opensources
MFEM: MFEM is an open-source library for solving partial differential equations using the finite element method. It is written in C++ and provides a wide range of features for solving both scalar and vector-valued problems, including heat transfer problems. MFEM provides several discretization options for solving heat transfer problems, including continuous and discontinuous Galerkin methods,…
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Boundary Conditions for Partial Differential Equations (1) Types of Boundaries
Here is a summary of the different types of boundary conditions that are commonly used in partial differential equation (PDE) problems: where $u$ is the variable (such as temperature or velocity) and $g$ is the fixed value specified by the boundary condition. where $u$ is the variable (such as temperature or velocity) and ${du/dn}$ is…
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NGSolve Tutorials (1) First NG Solve Example Part B
In NGSolve, the BilinearForm class represents a bilinear form defined on a finite element space. A bilinear form is a function that takes two finite element functions as input and returns a scalar. It is used to define the left-hand side of a weak formulation of a PDE. In the code you provided, a bilinear…
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NGSolve Tutorials (1) First NG Solve Example Part A
It is a example and explanation of which is described in docu.ngsolve.org Let us solve the Poisson problem of inding u satisfying in the unit square on the bottom and right parts of the boundary on the remaining boundary parts. The ngsolve module is a finite element library for solving partial differential equations (PDEs) in…
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Solving Partial Differential Equation by using Seperation of Variables : Part B. General Ways to Solve Wave Equation
The wave equation is a mathematical equation that describes the behavior of waves. It is a partial differential equation that describes how the wave’s amplitude, or strength, changes over time and space. The wave equation is used to model a wide range of phenomena in physics and engineering, including sound waves, light waves, and vibrating…
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Solving Partial Differential Equation by using Seperation of Variables : Part A. General Ways to Solve Laplace Equations
Here is another example of how to solve the Laplace equation for a simple rectangular domain using separation of variables, under the assumption of different boundary conditions: 1. Write the Laplace equation in the form: This is the general form of the Laplace equation, which holds for any rectangular domain. 2. Define the dimensions of…
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Partial Differential Equation (1) Laplace and Poisson Equation
From now, let’s talk about various partial differential equations. At first, I will talk about Laplace equation and Poisson equation. The Laplace equation is a second-order partial differential equation that arises in many physical systems, such as heat conduction, electrostatics, and fluid dynamics. It can be written in the following general form: where $u$ is…