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On Space-Time Quasiconcave Solutions of the Heat Equation
In the present paper, we will apply GRPS to a series of PDE with variable coefficients, including fourth-order parabolic equations, fractional heat equation, and fractional wave equation.
We, firstly, utilise the heat equation [u.sub.t] - [u.sub.xx] = 0 as an illustrative example.
The diffusion equation is sometimes called a heat equation or a continuity equation and it plays an important role in a number of fields of science.
Using the notation [u.sup.n](x) = u(x, [t.sup.n]) and a similar format as Crank-Nicolson (C-N) scheme to approximate the heat equation (1) with second order at the time moment [t.sup.n+1/2] = ([t.sup.n+1] + [t.sup.n])/2, we have
Ismailov, "An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions," Journal of Mathematical Analysis and Applications, vol.
The first type is the heat equation with operator coefficient.
Therefore, it seems interesting to handle the mixed fractional heat equation. In the recent paper of Xia and Yan , they introduced only the existence and uniqueness of the solution of a mixed fractional heat equation driven by a fractional Brownian sheet.
It is based on the heat equation in a solid region where the phase change interface is regarded as a moving boundary (Stefan, 1891).
The impulsive effects on blow-up and quenching were first studied by Chan and Deng , Chan, Ke and Vatsala  for a semilinear heat equation, and Chan and Kong  for a degenerate semilinear equation.
Since heat transfer coefficient is a boundary condition and an unknown variable in Equation (2), determining HTC from temperature data, which is the solution to the heat equation, requires a methodology to solve an inverse problem.

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