Solving Heat Problems Using Laplace Equation

Siti Aiyshah, Abdullah (2019) Solving Heat Problems Using Laplace Equation. [Final Year Project Report] (Unpublished)

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Abstract

Laplace’s equation is one of the particular important in applied mathematics because it is applicable to solve wide range of different physical and mathematical phenomena, including electromagnetism, fluid and solid mechanics, conductivity. We study on what method can be used to solve Laplace equation. In this study, we can see that how we use the method. There are few methods of Finite Difference schemes that can be used to solve Finite difference which are forward difference schemes, backward difference schemes, and central difference schemes. To solve its numerically also needs different methods and steps based on the solution we study. In this chapter, we can see that Finite Difference method being used to solve Laplace equation. We use central difference schemes to solve finite difference. We need to set our initial conditions and boundary conditions before we discretize Laplace equations to get the weak equations. The weak equations will be used in the code. In this chapter, we use central difference schemes and a few equations derived by using Taylor series expansion. Lastly, to solve elliptic Partial Differential Equations, we use Gauss Seidel method to solve the matrix problems. There are a few limitations on this study which are we did not get exactly result on graph. There are some errors to get exactly results and we need to try and error to get the answer. Based on that, we come out with conclusion that in this study, different values will effects the graph. Next is values used for boundary condition is not fixed and exactly right. It depends on the value that we give and see the graph. Different values of boundary condition well print different output. In future, we will develop a same model by improving our assumption and parameters that can represent the real life problem better in using this model to solve heat problems. This is because heat problems are universal that always be used anywhere and will effects by giving problems to other. So, we need to study and analyse more on needs to improve our model to be used.

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