Fundamental solution method for inverse source problem of plate equation

The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method （FSM） and the Tikhonov regularization technique axe used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.

EXPONENTIAL TIKHONOV REGULARIZATION METHOD FOR SOLVING AN INVERSE SOURCE PROBLEM OF TIME FRACTIONAL DIFFUSION EQUATION

In this paper,we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time.A novel regularization method,which we call the exponential Tikhonov regularization method with a parameter γ,is proposed to solve the inverse source problem,and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules.Whenγis less than or equal to zero,the optimal convergence rate can be achieved and it is independent of the value of γ.However,when γ is greater than zero,the optimal convergence rate depends on the value of γ which is related to the regularity of the unknown source.Finally,numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.

An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation

In this paper,an inverse source problem for the time-fractional diffusion equation is investigated.The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure.Here the sparsity is understood with respect to the pixel basis,i.e.,the source has a small support.By an elastic-net regularization method,this inverse source problem is formulated into an optimization problem and a semismooth Newton(SSN)algorithm is developed to solve it.A discretization strategy is applied in the numerical realization.Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR

In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.

MULTIPLICITY OF SOLUTIONS AND SOURCE TERMS IN A FOURTH ORDER NONLINEAR ELLIPTIC EQUATION

The authors investigatc relations between multiplicity of solutions and sourceterms of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition △2u＋c△u = bu+＋f inΩ, wherc Ω is a bounded open set in Rn with smoothbonndary and the nonlinearity bu+ crosses eigenvalues of △2 ＋c△. They investigate therelatiolls when the source term is constant and when it is generated by two eigenfuntions.

The enhanced volume source boundary point method for the calculation of acoustic radiation problem

The Volume Source Boundary Point Method (VSBPM) is greatly improved so that it will speed up the VSBPM's solution of the acoustic radiation problem caused by the vibrating body. The fundamental solution provided by Helmholtz equation is enforced in a weighted residual sense over a tetrahedron located on the normal line of the boundary node to replace the coefficient matrices of the system equation. Through the enhanced volume source boundary point analysis of various examples and the sound field of a vibrating rectangular box in a semi-anechoic chamber, it has revealed that the calculating speed of the EVSBPM is more than 10 times faster than that of the VSBPM while it works on the aspects of its calculating precision and stability, adaptation to geometric shape of vibrating body as well as its ability to overcome the non-uniqueness problem.

Radiogenic Source Identification for the Helium Production-Diffusion Equation

Knowledge of helium diffusion kinetics is critical for materials in which helium measurements are made,particulary for thermochronology.In most cases the helium ages were younger than expected,an observation attributes to diffusive loss of helium and the ejection of high energy alpha particles.Therefore it is important to accurately calculate the distribution of the source term within a sample.In this paper,the prediction of the helium concentrations as function of a spatially variable source term are considered.Both the forward and inverse solutions are presented.Under the assumption of radially symmetric geometry,an analytical solution is deduced based on the eigenfunction expansion.Two regularization methods,the Tikhonov regularization and the spectral cutoff regularization,are considered to obtain the regularized solution.Error estimates with optimal convergence order are shown between the exact solution and the regularized solution.Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods.

Regularization of the Inverse Problem for Time Fractional Pseudo-parabolic Equation with Non-local in Time Conditions

This paper is devoted to identifying an unknown source for a time-fractional diffusion equation in a general bounded domain.First,we prove the problem is non-well posed and the stability of the source function.Second,by using the Modified Fractional Landweber method,we present regularization solutions and show the convergence rate between regularization solutions and sought solution are given under a priori and a posteriori choice rules of the regularization parameter,respectively.Finally,we present an illustrative numerical example to test the results of our theory.

SOURCE TERM IDENTIFICATION WITH DISCONTINUOUS DUAL RECIPROCITY A PPROXIM ATION AND QUASI-NEWTON METHOD FROM BOUNDARY OBSERVATIONS

This paper deals with discontinuous dual reciprocity boundary element method for solving an inverse source problem.The aim of this work is to determine the source term in elliptic equations for nonhomogenous anisotropic media,where some additional boundary measurements are required.An equivalent formulation to the primary inverse problem is established based on the minimization of a functional cost,where a regularization term is employed to eliminate the oscillations of the noisy data.Moreover,an efficient algorithm is presented and tested for some numerical examples.