The Modified Equation Of High Order Approximate To Sideways Heat Equation

The inverse problems of mathematical physical problems is a hot research field in modern mathematics, the difficulty of this problem lies in its ill-posed. In this paper, we consider a class of inverse problems, i.e. Sideways Heat Equation. To be specific, we consider:For the above problems, in recent years, many mathematicians put for-ward many important results. For example, regularization, wavelet analysis and Fourier method, hyperbolic approximation, time discrete and so on. In this article, from hyperbolic approximation proposed by Weber and L.Elden, we introduce a new method of modified equation, i.e. consider the following equationSelect the appropriate parameters γ and κ, to approximate the original problem of the solution.Since there is no explicit description and proof of the optimal logarithmic error estimate of Sideways Heat Equation in the extant references. we give a brief discussion of the Worst Case Error according to a priori information in terms of||·||p which is the so-called’ stronger’norm.For this method, we shall employ the Fourier transform to give the the-oretic error estimate, from which it is easily to verify the optimal convergent rates with this method. In other words, we could achieve the logarithmic sta-bility according to a priori information in terms of||·||p, i.e., Theorem0.1:u is the exact equation of (1.1), v is the exact equation of (1.5) with random data. If||F||p≤M,p>0. while,||F||p=(f-∞∞(1+ω2)p(Fω)2dω)1/2. F(ω) is the Fourier transform of f(t)。If we choose while R[19] sat-isfies C1is constant。 We would also carry out the numerical experiments to show the validity and effectiveness of this method. Literally, the numerical results in our paper exhibit a sharp approximation to exact solution of the Sideways Heat Equation.