1 Introduction
Classical thermal conduction is based on the Fourier heat conduction law, where the heat flux is proportional to the temperature gradient. In this approximation, the rate of heat propagation is infinite. When considering heat transfer processes on small scales, we are forced to use more complex heat transfer models joseph1989heat ; straughan2011heat . A classic example of nonFourier heat conduction equations is the hyperbolic heat conduction model when the effects of heat flux relaxation are taken into account. The mathematical model becomes more complicated but remains local: we have a secondorder evolution equation for temperature or a system of firstorder equations for temperature and heat flux. In the general case, we should focus on nonlocal heat transfer models, where the thermal state depends on prehistory. Memory effects of heat flow and heat capacity are accounted for using nonlocal evolution equations for temperature gurtin1968general ; nunziato1971heat with integral terms with differencetype kernels.
The problems of numerical solution of nonlocal in time IBVPs are well known. We can use some or other quadrature formulas to approximate the integral time terms and have suitable properties on stability and convergence of the approximate solution to the exact one (see, for example, ChenBook1998
). The main problems are related to the computational cost of finding the solution on the new time level. We have to operate with approximate solutions for all prior time moments. Because of this, we are particularly interested in approaches that allow us to go from a nonlocal model to a local model not much more complex than the classical heat conduction equation.
We use in our work a wellknown method of approximate solution of the Volterra integral equations linz1985analytical , which is associated with an approximation of the kernel by a sum of exponents. The construction of such approximations is an independent nonlinear approximation problem braess1986nonlinear . The justification of such a transition for evolutionary firstorder integrodifferential equations with solution memory and solution derivative in time is given in our works vabMemory ; vabMemoryB . The present paper considers evolution equations for heat conduction problems with heat flow memory and heat capacity memory.
The paper is organized as follows. In Section 2, we describe nonlocal heat conduction problems with heat flow memory and heat capacity memory. In Section 3, we discuss the transition from a nonlocal problem with heat flow and energy distribution functions as a sum of exponents to the Cauchy problem for a local system of firstorder evolution equations. Section 4 is devoted to the construction and study of twolevel operatordifference schemes. In Section 5, we present examples of numerical solutions of onedimensional spatial heat conduction problems with memory. The final section is the conclusion.
2 Nonlocal heat conduction problems
We limit ourselves to considering linear models of heat conduction of a solid isotropic body. In classical heat conduction theory, the heat flow at a point at some time is determined by the temperature gradient (Fourier’s law of heat conduction) when
(2.1) 
where is the thermal conductivity. For a rigid conductor the internal energy equation is
(2.2) 
where is the distributed heat suction. The energy increments are due to changes in temperature, so that
(2.3) 
where is the heat capacity. Substituting (2.1), (2.3) into (2.2) gives the heat conduction equation
(2.4) 
Mathematical models of heat conduction with memory are proposed in gurtin1968general ; nunziato1971heat . The prehistory of the process is accounted in gurtin1968general by specifying the heat flow by the relation
(2.5) 
where is a positive, decreasing function relaxation of the heat flux. In the special case ( is a function), the Fourier law follows from (2.1), (2.5). The relation (2.5) leads us instead of the equation (2.4) to the evolutionary integrodifferential equation
(2.6) 
Let’s assume that
(2.7) 
where is the relaxation time. In this case, from (2.5) we have the MaxwellCattaneo law cattaneo1948sulla
(2.8) 
for the fluxes. This leads us to the hyperbolic heat conduction equation bubnov1976wave
(2.9) 
The fundamental point is that instead of a nonlocal model (2.5) for the exponential dependence of the relaxation function on time (2.7), we have a local model (at each individual point ) for the dependence of heat flux on temperature gradient as a differential relation (2.8). Instead of the integrodifferential equation (2.6), a secondorder evolution equation (2.9) is used.
For the heat flux relaxation function we will use nunziato1971heat the expression
(2.10) 
Similar relations are used for internal energy relaxation when
(2.11) 
(2.12) 
From (2.2), (2.5), (2.10)–(2.12), we obtain the heat conduction equation with memory
(2.13) 
We consider a numerical solution of the boundary value problem for the equation (2.13). For simplicity, we restrict ourselves to the case of a bounded domain with a boundary on which homogeneous Dirichlet boundary conditions are given:
(2.14) 
The initial thermal state is defined according to
(2.15) 
Singlevalued solvability of boundary value problems of type (2.13)–(2.15) in the corresponding functional classes, the qualitative properties of the solutions are discussed, for example, in davis1976hyperbolicity ; davis1978linear .
The object of our consideration is computational algorithms for approximate solutions of heat conduction problems with allowance for memory effects. The main problems are generated by the nonlocality of the equation (2.13), the fact that the solution depends on the complete prehistory of the process. When finding the solution at a new level in time, we have to consider the solution at previous levels in time, which, in particular, is associated with high computational and memory costs. Distinct possibilities of a principal reduction of computational work are seen on the example of the transition to the local equation (2.9) due to transition from the integrodifferential relation (2.5) to the differential relation (2.8) when the heat flow distribution function is given as (2.7). We can expect a similar simplification of the problem when the heat flux and energy distribution functions are represented as
(2.16) 
For general dependencies this representation corresponds to using nonlinear approximations of functions by the sum of exponents braess1986nonlinear .
3 Local system of equations
After a finiteelement or finitevolume approximation over the space, we consider the discrete problem in the corresponding finitedimensional Hilbert space KnabnerAngermann2003 ; QuarteroniValli1994 . For the scalar product and norm we use the notation and for respectively. A positively determined selfadjoint operator is associated with the Hilbert space , in which , .
The dimensionless problem (2.13)–(2.15) leads us to the equation
(3.1) 
for . The operator is associated with an approximation of the Laplace operator for functions satisfying boundary conditions (2.14). It is natural to assume that in it is constant (independent of ) and . We consider the Cauchy problem for the Volterra evolutionary equation (3.1) when we use the initial condition
(3.2) 
The transition from a nonlocal to a local problem when approximating the differencetype kernel by a sum of exponents is provided by introducing new desired variables. For the problem (2.16), (3.1), (3.2), we put
For these auxiliary functions, we have the local relations
(3.3) 
(3.4) 
Equation (3.1) is written as
Given (3.4), we have
(3.5) 
The initial condition (3.2) for is supplemented by initial conditions for auxiliary functions:
(3.6) 
Thus from the nonlocal problem (2.16), (3.1), (3.2) for , we arrive at the local Cauchy problem for the system of equations (3.2)–(3.6).
The main result concerns the stability for the initial data and the righthand side for the system of equations (3.3)–(3.5). The correctness of this linear problem is established under the assumption that the coefficients in the representations (2.16) are positive:
(3.7) 
Theorem 1.
Proof.
4 Twolevel difference scheme
When approximating firstorder evolution equations, it is natural to focus on standard twolevel schemes with weights LeVeque2007 ; SamarskiiTheory . Without limiting generality, we will assume that the time grid is uniform, i.e. where is time step. We denote the approximate solution at time by and let
where is the weight parameter.
For the approximate solution of the problem (3.2)–(3.6), we will use a twolevel scheme
(4.1) 
(4.2) 
(4.3) 
(4.4) 
where, for example, . The stability of this scheme is established under standard constraints on the weight of .
Theorem 2.
Proof.
If we multiply equation (4.1) by , equation (4.2) by , and equation (4.3) by , we get the inequality
(4.6) 
Given that
at , we have
From the inequality (4.6), it follows
(4.7) 
Under our constraints on , we have
Thus from the inequality (4.7), we obtain the estimate
From this follows the inequality being proved (4.5). ∎
The computational realization of the scheme with weights (4.1)–(4.4) can be organized as follows. From equations (4.2), (4.3), we have
(4.8) 
(4.9) 
Substituting expressions (4.8) and (4.9) into equation (4.1) gives
(4.10) 
where is a unit operator, and
For the righthand side of equation (4.10), we have
The operator , where , and so equation (4.10) is uniquely solvable for . After finding , the auxiliary quantities are calculated according to (4.8), (4.9). The increase in the computational complexity of the numerical solution of problems with memory is not fundamental in comparison to the problem without memory effects.
5 Numerical examples
In the above calculations, we have limited ourselves to onedimensional problems in space. Considering the current state of numerical solution of boundary value problems for elliptic equations, the transition to more complex multidimensional problems of heat conduction with memory is not of fundamental nature. The methodological issues of estimating the rate of convergence of the approximate solution to the exact solution of the system of evolution equations are not discussed. We present numerical data on sufficiently detailed computational grids in time and space when we can neglect computational errors.
To illustrate the peculiarities of problems with memory effects, we consider the equation (3.1) in which
i.e., in (2.6) . The system of equations (3.3)–(3.5) takes the form
(5.1) 
(5.2) 
(5.3) 
For (5.1)–(5.3), a Cauchy problem is posed when
(5.4) 
It makes sense to distinguish the following separate classes of heat conduction problems, for which we will give the corresponding equation for temperature.
 1. Classical thermal conductivity.

The standard model of thermal conductivity without regard to memory effects corresponds to the initial condition (3.2) and :
 2. Heat flux memory
 3. Heat capacity memory
 4. General case
Numerical illustrations of the features of local generalized thermal conductivity models are given, usually (see, for example, shen2008notable ; hu2009study ; zhang2013numerical ) tracing the dynamics of the initial temperature profile. We consider a similar test problem whose numerical solution is carried out with a difference space approximation. Let us assume that in dimensionless variables and a uniform grid with step is introduced. Let us denote by the set of internal nodes of the grid in space. Given (2.14), for , we define the difference operator by the relation
The initial condition (3.2) is taken as
The calculations were performed on a grid over a space with , with , and . A purely implicit scheme was used ( in the scheme (4.1)–(4.4)).
The dynamics of the process when using model 1 is presented in Fig. 1, and for model 4 — in Fig. 2. The effects of heat flux memory and heat capacity lead to a significant rearrangement of the temperature profile. The influence of individual parameters of the mathematical model is traced to models 2 and 3.
6 Conclusions

A boundary value problem for the first order integrodifferential evolution equation with differencetype kernels, which describes heat conduction processes in solids in linear approximation with allowance for memory effects, is formulated. Processes with a memory of heat flow and memory of heat capacity (internal energy) are modeled.

The nonlocal time problem is reduced to a system of local evolution equations of the first order under the condition that the heat flow and heat capacity relaxation functions are represented as a sum of exponents. We obtained an estimate of the stability of the solution of the Cauchy problem for the system of equations concerning the initial data and the righthand side in the corresponding Hilbert spaces.

A unconditionally stable scheme with weights for an evolutionary system of equations modeling thermal conductivity in solid media with memory has been proposed and investigated. The transition to a new level in time is not more complicated than the computational implementation of standard schemes with weights for the classical heat conduction equation.

Numerical results of using a purely implicit scheme in approximate solutions of heat conduction problems with allowance for memory effects are presented. The evolution of the initial state at different values of model parameters is considered.
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