We employ a quasi‐boundary regularization to construct a two‐point boundary value problem for multi‐dimensional backward heat conduction equations. The multidimensional backward heat conduction problem (BHCP) is renowned as severely ill‐posed because the solution does not fullly depend on the data. In order to numerically tackle the multi‐dimensional BHCP, we propose a Lie‐group shooting method (LGSM) in the time direction to find the unknown initial conditions. The pivot point is based on the establishment of a one‐step Lie group element G(T) and the construction of a generalized mid‐point Lie group element G(r). Then, by imposing G(T) = G(r) we can search for the missing initial conditions through a minimum discrepancy to the real targets by the numerical ones, in terms of the weighting factor r ? (0, 1). When numerical examples are tested, we find that the LGSM is applicable to the BHCP. Even with noisy final data, the LGSM is also robust against disturbance.

This article is aimed to reconstruct a time-dependent heat source for a one-dimensional heat conduction equation. The extra measurement data are used to transform the original equation into a homogeneous equation with three-point boundary conditions. Then the modified polynomial expansion method is developed to deal with the resulting three-point boundary-value problem. By considering the characteristic length, the modified polynomial expansion method can obtain a convergent series solution and improve the stability of the algorithm. The accuracy and efficiency of the present method are validated by comparing the estimating results with those of designed examples even under noisy measurement data.

For a second-order nonlinear singularly perturbed boundary value problem (SPBVP), we develop two novel algorithms to find the solution, which automatically satisfies the Robin boundary conditions. For the highly singular nonlinear SPBVP the Robin boundary functions are hard to be fulfilled exactly. In the paper we first introduce the new idea of boundary shape function (BSF), whose existence is proven and it can automatically satisfy the Robin boundary conditions. In the BSF, there exists a free function, which leaves us a chance to develop new algorithms by adopting two different roles of the free function. In the first type algorithm we let the free functions be the exponential type bases endowed with fractional powers, which not only satisfy the Robin boundary conditions automatically, but also can capture the singular behavior to find accurate numerical solution by a simple collocation technique. In the second type algorithm we let the BSF be solution and the free function be another variable, such that we can transform the boundary value problem to an initial value problem (IVP) for the new variable, which can quickly find accurate solution for the nonlinear SPBVP through a few iterations.

We consider an inverse problem of a nonlinear heat conduction equation for recovering unknown space-dependent heat source and initial condition under Cauchy-type boundary conditions, which is known as a sideways heat equation. With the aid of two extra measurements of temperature and heat flux which are being polluted by noisy disturbances, we can develop a Lie-group differential algebraic equations (LGDAE) method to solve the resulting differential algebraic equations, and to quickly recover the unknown heat source and initial condition simultaneously. Also, we provide a simple LGDAE method, without needing extra measurement of heat flux, to recover the above two unknown functions. The estimated results are quite promising and robust enough against large random noise.

In this paper, we propose a semi‐analytical method to deal with the backward heat conduction problem due to a quasi‐boundary idea. First of all, the Fourier series expansion technique is used to calculate the temperature field u(x, t) at any time t < T. Second, we consider a direct regularization by adding the term αu(x, 0) into the final time condition to obtain a second kind Fredholm integral equation for u(x, 0). The termwise separable property of the kernel function allows us to transform the backward problem into a two‐point boundary value problem and therefore, a closed‐form solution is derived. The uniform convergence and error estimation of the regularized solution u^α (x, t) are provided and a tactic to choose the regularization parameter is suggested. When several numerical examples are amenable, we discover that the present approach can retrieve all the past data very well and is robust even for seriously noised final data.