The MAFS For Solving Laplace And Biharmonic Equations With Non-harmonic And Non-biharmonic Boundary Conditions

Partial Differential Equations(PDEs)are significant in modeling physical phenomena.They have applications in physicals,engineering,mathematics,and finance.Unfortunately,only a small number of PDEs have known analytical solutions.Therefore,being able to efficiently obtain the accurate numerical solutions is crucial.The method of fundamental solutions(MFS)has been known as an effective boundary meshless method for solving homogeneous differential equations with smooth boundary conditions and boundary shapes.The two features that have made MFS so popular are,primarily,its simplicity and the ease with which it can be implemented.However,despite this,there are still some important issues associated with the method which have not yet been satisfactorily addressed.One of these issues is how to appropriately choose the location of the source in the MFS.Especially for the problems whose boundary with sharp corners.In this thesis,we revisit another powerful boundary method,the method of approximate fundamental solutions(MAFS).It is similar to the method of fundamental solutions but it is based on the use of Delta-shaped basis functions which satisfy the majority of the boundary conditions of the problem considered.The MAFS first suggested for solving elliptic problems and heat equations in domains with fixed boundaries.In the MAFS,the approximate fundamental solutions for various governed equations can be easily constructed.The placement of the source points is also simple.The main content of this thesis is as follows:1.Introducing the method of fundamental solution and how to use it to solve some boundary problems;2.Introducing Delta-shaped basis function,giving the example for constructing the approximate fundamental solutions,constructing the approximate fundamental solution for various differential operator,introducing regional transformation,knowing how to transform a non-regular region to a regular domain and how to use the MAFS to solve some boundary problems;3.In this thesis,we will apply the MAFS for solving the Laplace equation with non-harmonic boundary conditions and the Biharmonic equation with non-biharmonic boundary conditions with highly irregular or non-smooth domains.We will compare the performance of the MAFS and the MFS in these types of problems.