Numerical Solution Of2-D Heat Conduction Equation And Fractional Order Differential Equations By Using Wavelets

In recent years, the heat conduction equation and the fractional order differentialequation are widely applied in many fields of science,for example in the heat spread ofelectromagnetic fields,viscoelastic mechanics,the diffusion theories and so on.Manynumercal methodes of solving these equations also arises. Mathematicians are so keento solve integral-differential equations by using wavelet analysis at present.Theapplication of wavelet analysis are so widely,which entirely due to the originality andthe integrity of its mathematical mechanism. Compared with many other disciplinessuch as functional analysis, harmonic analysis, approximation theory, numericalanalysis and so on,Wavelet analysis which is the crystal of their perfect combination isa complete theoretical system. The method that numerical solution of fractionaldifferential equation and heat transfer equation by using wavelet has become a new hottopic.The paper firstly introduces the development history of wavelet analysis, heatconduction equations and some work about fractional differential equations at present.Then we give the definitions and the properties of wavelets, some prior knowledge offractional differential equations and the heat conduction equations.Secondly, the paper mainly research methods of solving the initial-boundaryvalue problems of2-D heat conduction equation by using the properties of waveletsbases of Hermite Cubic Splines.Approximation of the unknown function is limited.Sothe original problem is translated into Sylvester equation. and it overcomes thedifficulties that instability of the numerical solution because of time variable. Thenumerical example shows that the method is effective and accurate.Finally, the dissertation considers numerical solution of fractional differentialequations by using Sine-cosine wavelets or Haar wavelet resperctively.For solving thelinear fractional differential equation,we convert it into Volterra integral equation, andthen solve it by using Sine-cosine wavelets. Numerical example gives the verification.At last we use orthogonal Haar wavelet to solve nonlinear fractional differential equation because of its properties such as compact support, orthogonality and so on.We can obtain non-linear algebraic equations, and verify the efficiency of the methodby the numerica example.