Numerical Computation Of Fractional Constitutive Model For Potential And Non-stationary Heat Transfer Problems

In recent decades,the fractional calculus theory has gradually attracted the attention of researchers and developed rapidly.In comparison with the mathematical models of integer-order calculus,the models involving fractional derivative provides a more adequate and accurate way to interpret the properties such as memory of complex environment,heredity,non-locality,self-similarity,long-range-dependence and so on.Due to the difficulty and complexity of fractional calculus to obtain analytical solutions,numerical techniques are needed to use.This paper aims to numerically solve the integer-order constitutive models of potential and non-stationary heat transfer problems,and the authors solve them by building the fractional constitutive models.This model was applicable not only to the generalized fractional order,but also to the integer order model given in this paper.All text examples given in this paper are based on actual problems to abstract a general mathematical model so as to solve it with the given numerical method.In this paper,the main points are divided into the following sections.?1?we discuss the Poisson and Laplace equations of two-dimensional potential problems together with the Dirichlet and Neumann boundary conditions.Firstly,we introduce the definitions of two-dimensional Block-Pulse functions?BPFs?,and construct the orthogonal basis vector of BPFs.Then the solution function of this problem is approximately represented by the orthogonal basis vector,and the derivative term of original problem is written in a vector form.Lastly,we solve the linear system by dispersing the unknown variables.The numerical results show that our method is easy to construct and less time-consuming.Moreover,a satisfactory numerical precision can be achieved by truncating the appropriate numbers of series terms.?2?we firstly utilize the operational matrix method to obtain the numerical solutions of Laplace and Poisson equations of three-dimensional potential problems.The proposed method is based on the one-dimensional differential operational matrix of BPFs to construct the three-dimensional differential operational matrix of BPFs.Then,each term of this original problem combined with its boundary condition is approximately represented in a vector form.Lastly,we obtain the numerical solutions by solving this system.The method in this chapter differs from the previous methods of using spherical harmonic functions and three-dimensional Taylor series for the three-dimensional potential problems.Our method is based on the three-dimensional Block-Pulse orthogonal functions,and it runs faster.Moreover,when the series terms are expanded into 64,the obtained numerical precision can achieve10^-310^-4.?3?we apply the second kind of Chebyshev wavelet method to solve one-dimensional non-steady heat transfer model with constant coefficients.The proposed method is based on the definition of the second kind of Chebyshev wavelet combined with its operational matrix of fractional integration.Then,the operational matrix of fractional-order integration is utilized to transform the original problem into a system of linear algebraic equations.We solve it and obtain the numerical solutions.The wavelets method can accurately exert the details of short-term load sequence compared with the traditional Fourier analysis,so it can achieve higher numerical precision.Lastly,the obtained numerical solutions show that our method is effective and feasible.?4?we derive the numerical solutions of one-dimensional non-steady heat transfer problem with variable coefficients based on the Chebyshev polynomials.The convergence precision can achieve10^-910^-10by using orthogonal polynomials functions to approximate the solution function to the polynomials types.This chapter aims to deal with the nonlinear problem with variable coefficients.By introducing the product operational matrix,the original problem is transformed into a unified vector form.In addition,the error analysis is investigated and the obtained numerical results manifest our algorithm can achieve a higher convergence precision for solving this kind of problem.