Abstract
In this paper, we consider the nonlinear inverse-time heat problem with a conformable derivative concerning the time variable. This problem is severely ill posed. A new method on the modified integral equation based on two regularization parameters is proposed to regularize this problem. Numerical results are presented to illustrate the efficiency of the proposed method.
Highlights
Partial differential equations (PDEs) with different types of boundary conditions play an essential tool in modelling natural phenomena
An example is the backward heat problem (BHP) where the goal is to recover the previous status of a physical field from the present information
Fu et al [3] applied a wavelet dual least square method to investigate a BHP with constant coefficients, in [4], Hao et al gave an approximation for this problem using a nonlocal boundary value problem method, Hao and Duc [5] used the Tikhonov regularization method to give an approximation for this problem in a Banach space, and Tautenhahn in [6] established an optimal error estimate for a backward heat equation with constant coefficients
Summary
Partial differential equations (PDEs) with different types of boundary conditions play an essential tool in modelling natural phenomena. Using the stabilized quasireversibility method, the final value problem for a class of nonlinear parabolic equations is investigated by Trong and Tuan [7], and in [8], the authors used the integral equation method to regularize the backward heat conduction problem and they obtained some error estimates. We use the following modified integral equation to approximate or to regularize the solution of problems (1)–(3): uε,c(x, t). The term exp(kn(bα − tα)/α) is replaced by a stable term which depends on two regularization parameters defined by (εkn + e− kn((bα+c)/α))((tα− bα)/(bα+c)), where the first one (ε) captures the measuring error and the second one (c) captures the regularity of the solution. Erefore, in this paper, we shall use integral equation (6) to approximate or to regularize problems (1)–(3)
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