the method of central difference for the inverse time dependent heat source problem abstract this paper investigates the inverse problem of determining a heat source in a parabolic equation where data are given at some fixed location this problem is ill posed i e the solution if it exists does not depend continuously on the data a central difference regularization method is given and an order optimal stability estimate is obtained numerical results for several benchmark test problems indicate that the central difference regularization method is an accurate and flexible method to determine the unknown time dependent heat source

the mollification method and the numerical solution of the inverse heat conduction problem by finite differences abstract the inverse heat conduction problem involves the calculation of surface heat flux and or temperature histories from transient measured temperatures inside solids we consider the one dimensional semi infinite linear case and present a new solution algorithm based on a data filtering interpretation of the mollification method that automatically determines the radius of mollification depending on the amount of noise in the data and finite differences a fully explicit and stable space marching scheme is developed we describe several numerical experiments of interest showing that the new procedure is accurate and stable with respect to perturbations in the data even for small dimensionless time steps

a fast finite difference method for distributed order space fractional partial differential equations on convex domains abstract fractional partial differential equations pdes provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long range spatial interactions which cannot be modeled accurately by classical second order diffusion equations however numerical methods for space fractional pdes usually generate dense or full stiffness matrices for which a direct solver requires o n 3 computations per time step and o n 2 memory where n is the number of unknowns the significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three dimensional space fractional diffusion equations computationally intractable fast numerical methods were previously developed for space fractional pdes on multidimensional rectangular domains without resorting to lossy compression but rather via the exploration of the tensor product form of the toeplitz like decompositions of the stiffness matrices in this paper we develop a fast finite difference method for distributed order space fractional pdes on a general convex domain in multiple space dimensions the fast method has an optimal order storage requirement and almost linear computational complexity without any lossy compression numerical experiments show the utility of the method

a mollified space marching finite differences algorithm for the inverse heat conduction problem with slab symmetry abstract the one dimensional inverse heat conduction problem ihcp for a slab is considered a new solution algorithm based on a data filtering interpretation of the mollification method is presented and a fully explicit space marching finite differences scheme is developed after showing the numerical stability of the algorithm the efficiency of the method is demonstrated by means of several examples the numerical results show the procedure to be very useful in the presence of noisy data even for some nonlinear cases

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

determining surface temperatures from interior observations we consider the inverse heat transfer problem for the case of the one dimensional heat equation in a quarter plane and present a new solution algorithm together with error bounds of logarithmic convexity type our method is based on regularizing the initial value problem for the heat equation run sideways we describe a numerical experiment with rapidly varying temperature profiles and explain how the method can be extended to problems with temperature dependent thermal properties

identifying an unknown source in time fractional diffusion equation by a truncation method in this paper an inverse source problem for the time fractional diffusion equation is investigated we prove a conditional stability for this problem a truncation method is presented to deal with the ill posedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter numerical examples show that the proposed method is effective and stable

a new regularization method for solving a time fractional inverse diffusion problem in this paper we consider an inverse problem for a time fractional diffusion equation in a one dimensional semi infinite domain the temperature and heat flux are sought from a measured temperature history at a fixed location inside the body we show that such problem is severely ill posed and further apply a new regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under the a priori bound assumptions for the exact solution finally numerical examples are given to show that the proposed numerical method is effective

an inverse problem for fractional diffusion equation in 2 dimensional case stability analysis and regularization abstract in this paper we investigate an inverse problem for a time fractional diffusion equation which is highly ill posed in the two dimensional setting based on an a priori assumption we give a conditional stability result some new regularization methods are constructed for solving the inverse problem and the corresponding error estimates are proved for numerical illustration several examples are constructed to demonstrate the feasibility and efficiency of the proposed methods

finite volume method for solving a one dimensional parabolic inverse problem in this paper finite volume method is used to solve a one dimensional parabolic inverse problem with source term and neumann boundary conditions for the first time some advantages of this approach are developing difference schemes and maintaining certain properties of the physics of the problems especially for the treatment of the source term and the unknown boundary conditions numerical results show that our method is more effective

spectral regularization method for solving a time fractional inverse diffusion problem abstract in this paper we consider an inverse problem for a time fractional diffusion equation with one dimensional semi infinite domain the temperature and heat flux are sought from a measured temperature history at a fixed location inside the body we show that such problem is severely ill posed and further apply a spectral regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under a priori bound assumptions for the exact solution finally numerical examples are given to show that the proposed numerical method is effective

invariants nilpotent orbits and prehomogeneous vector spaces provisionally in the introduction we call such a triple g o ad g i i 0 a prehomogeneous vector space of vinberg type abbreviated pv of v type the purpose of this paper is to construct relative invariants of pv of v type in a unified way let us explain the way of construction of invariants we may assume that the original graded lie algebra g is zh graded i e g ie 2l g i we may also assume that i 2 without loss of generality cf 1 2 4 let p v be a representation of g and

fourier truncation method for high order numerical derivatives we consider a classical ill posed problem numerical differentiation with a new method we propose fourier truncation method to compute high order numerical derivatives a holder type stability estimate is obtained a numerical implementation is described numerical examples show that the proposed method is effective and stable

sideways heat equation and wavelets the sideways heat equation is considered as a model ill posed problem the aim of this paper is to indicate a possibility of a new approach for solving this problem based on an application of wavelet basis decomposition of measured data the meyer wavelets are applied to formulate a regularized solution which is convergent to the exact one when a data error tends to zero

two regularization methods and the order optimal error estimates for a sideways parabolic equation in this paper we consider an inverse heat conduction problem which appears in some applied subjects this problem is ill posed in the sense that the solution if it exists does not depend continuously on the data a tikhonov type s regularization method and a fourier regularization method are applied to formulate regularized solutions which are stably convergent to the exact ones with order optimal error estimates a numerical example shows that the computational effect of these methods are all satisfactory

a regularization framework for mildly ill posed problems connected with pseudo differential operator abstract recently filter based regularization methods have been well investigated for ill posed problems when the forward operators are compact there are many ill posed problems connected with pseudo differential operators but there is no uniform method for this kind of problems the work on generalization of filter based regularization methods to pseudo differential operator is necessary in this paper we present a regularization framework for solving the mildly ill posed problems involved pseudo differential operators a general regularization method for this kind of problems is given the order optimal error estimates are derived under the usual source conditions as an example a new fractional tikhonov regularization method could be cast into the general framework numerical experiments are conducted for showing the validity of the new fractional tikhonov method

optimal error bound and fourier regularization for identifying an unknown source in the heat equation in this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation under an a priori condition we answer the question concerning the best possible accuracy for the problem the fourier regularization method is utilized for solving the problem and its convergent rate is analyzed numerical results are presented to illustrate the accuracy and efficiency of the method

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

wavelet calculus and finite difference operators this paper shows that the naturally induced discrete differentiation operators induced from a wavelet galerkin finite dimensional approximation to a standard function space approximates differentiation with an error of order 0 h2d 2 where d is the degree of the wavelet system the degree of a wavelet system is defined as one less than the degree of the lowest order nonvanishing moment of the fundamental wavelet we consider in this paper compactly supported wavelets of the type introduced by daubechies in 1988 the induced differentiation operators are described in terms of connection coefficients which are intrinsically defined functional invariants of the wavelet system defined as l2 inner products of derivatives of wavelet basis functions with the basis functions themselves these connection coefficients can be explicitly computed without quadrature and they themselves have key moment vanishing properties proved in this paper which are dependent upon the degree of the wavelet system this is the basis for the proof of the principal results concerning the degree of approximation of the differentiation operator by the wavelet galerkin discrete differentiation operator

central difference schemes in time and error estimate on a non standard inverse heat conduction problem inverse heat conduction problems ihcp are severely ill posed in the sense that the solution if it exists does not depend continuously on the data but now the results available in the literature on ihcp are mainly devoted to the standard sideways heat equation this paper remedies this by a central difference schemes in time which itself has a regularizing effect for a non standard ihcp which appears in some applied subjects an error estimate is obtained and the error estimate also gives information about how to choose the step length in the time discretization a numerical example shows that the computational effect of this method is satisfactory

a stable space marching finite differences algorithm for the inverse heat conduction problem with no initial filtering procedure abstract we consider the one dimensional semi infinite linear inverse heat conduction problem ihcp and present a new solution algorithm based on singular perturbation techniques the new method successfully incorporates the needed initial filtering procedure of the noisy data into the marching scheme the concept of formal stability for the ihcp is analyzed and its relationship with the high frequency components of the measured data funciton is further studied by comparing the new method with the mollification method and the method of weber a fully explicit space marching finite differences scheme is developed and several numerical experiments of interest showing the accuracy and stability properties of the algorithm in the presence of noisy data and for small dimensionless time steps are investigated