A Method Of Fundamental Solutions Based On Geodesic Distance For Inhomogeneous Anisotropic Heat Conduction Problem And Its Inverse Problems

Heat conduction problem and its inverse problems arise in many scientific andengineering applications. The homogeneous problems have been extensively studiedin the past years. Unfortunately, the studies of inhomogeneous problems are evenmore difficult due to their complexity. This thesis focuses on inhomogeneous heatconduction problem and its inverse problems in anisotropic materials which are ofgreat importance in practice.In Chapter 2, we will solve inhomogeneous anisotropic IHCP (inverse heatconduction problem), in which the main difficulty is how to obtain a particularsolution of the governing equation. For this IHCP, a method of combing the method ofradial basis function (RBF) Multiquadric (MQ) and the method of fundamentalsolutions (MFS), which are both based on geodesic distance, is proposed andemployed. We consider the problem in two conditions according to whether theinhomogeneous term is related to the time t. When the inhomogeneous term isindependent of t, at first, to approximate a particular solution of the governingequation by a linear combination of RBF MQ based on geodesic distance, in whichthe combinatorial coefficients are constant, then to calculate the coefficients throughsome collocation points, at last, to solve the corresponding homogeneous problemusing the method of fundamental solutions based on geodesic distance. When theinhomogeneous term is related to t, at first, to approximate a particular solution ofthe governing equation by a linear combination of RBF MQ based on geodesicdistance, in which the combinatorial coefficients are related to t, then to obtain anone order ordinary differential equation of the coefficients' vector related to tthrough some collocation points, and then to calculate the coefficients at any time through approximating the differential coefficient by forward difference and choosingthe beginning time value, at last, to solve the corresponding homogeneous problemusing the method of fundamental solutions based on geodesic distance. Theinterpolation matrixes arising from the RBF MQ and MFS are highly ill-conditioned,on the other hand, the IHCP is highly ill-posed. Thus, a regularization method shouldbe employed. In this thesis, we choose truncated singular value decomposition (TSVD)to solve the resulting matrix equations, while the regularization parameter of TSVD isdetermined by the L-curve criterion. In the end, several numerical examples arepresented to certify the method with both exact and noisy data. Meanwhile theconvergency of the method and stability with respect to the data noise, and therelationship between the numerical results and the parameter T and c are alsoanalyzed.In Chapter 3, we will solve inhomogeneous anisotropic BHCP (backward heatconduction problem). The method is similar to the one in Chapter 2. The onlydifference is to calculate the coefficients at any time through approximating thedifferential coefficient by backward difference and choosing the final time value whenthe inhomogeneous term is related to t and after obtaining the ordinary differentialequation of the coefficients' vector. In the end, several numerical examples are alsopresented to certify the efficiency of the proposed method. The relationship betweenthe final time and the numerical solution is analyzed in addition.In Chapter 4, we will solve inhomogeneous anisotropic heat conduction equation.The method is the same as the one in Chapter 2, only extending it for any dimensionalspace. Similarly, numerical experiments are performed to demonstrate the efficiencyand superiority of the method for piecewise smooth domain. In addition, the effect ofthe number of the collocation points is discussed.