Academic buildings in a typical university campus occupy 42% of the total space and are responsible for nearly 50 percent of the total energy use and carbon emissions of the campus. Forecasting of energy consumption in this energy intensive building category could help higher education institutions in taking energy saving initiatives and in revising their building operating strategies. Reliable predictive techniques does not only help in forecasting a building’ energy consumption, but also help in identifying a variety of factors affecting the energy consumption of that building. This study attempts to forecast and benchmark the daily electricity consumption of an academic building situated in London, United Kingdom using two different data-driven modeling techniques, i.e., Multiple Regression and Artificial Neural Network. Hourly dataset for the electricity consumption was available for the period 2007 to 2011 from the smart meter whereas hourly data of different factors such as ambient temperature, relative humidity, wind speed and solar radiation were downloaded from the website of environmental research group of Kings College London. The performances of the two predictive models have been critically analyzed by comparing their predicted consumption with a real dataset of the same building for the year 2012. A comparison shows that both Multiple Regression (MR) and Artificial Neural Network (ANN) perform reasonably well with a Mean Absolute Percentage Error (MAPE) of 3.34% and 2.44% for working days and 5.12% and 4.59% for non-working days respectively. ANN performs slightly better than MR. This energy consumption forecasting approach can easily be adapted for predicting energy use of similar buildings.
1.
Introduction
We are concerned with Atangana-Baleanu variable order fractional problems:
where 0<α(x)<1, ABCDα(x)(x) denotes the α(x) order Atangana-Baleanu Caputo derivatives, B(u) is the linear boundary condition, which includes initial value condition, periodic condition, final value condition and so on.
The α(x)(0<α(x)<1) order Atangana-Baleanu Caputo derivatives of a function u(x) is firstly defined by Atangana and Baleanu [1]
where Eα(x)(x) is the Mittag-Leffler function.
Fractional order differential equations (FDEs) have important applications in several fields such as materials, chemistry transmission dynamics, optimal control and engineering [2,3,4,5,6]. In fact, the classical fractional derivatives are defined with weak singular kernels and the solutions of FDEs inherit the weak singularity. The Mittag-Leffler (ML) function was firstly introduced by Magnus Gösta Mittag-Leffler. Recently, it is found that this function has close relation to FDEs arising in real applications.
Atangana and Baleanu [1] introduced a new fractional derivative by using the ML function, which is nonlocal and nonsingular. The new fractional derivatives is very important and have been applied to several different fields (see e.g. [7,8,9]). Up to now, several numerical algorithms have been developed for solving Atangana-Baleanu FDEs. Akgül et al. [10,11,12] proposed effective difference techniques and kernels based approaches for Atangana-Baleanu FDEs. On the basis of the Sobolev kernel functions, Arqub et al. [13,14,15,16,17] proposed the numerical techniques for Atangana-Baleanu fractional Riccati and Bernoulli equations, Bagley-Torvik and Painlev equations, Volterra and Fredholm integro-differential equations. Yadav et al. [18] introduced the numerical algorithms and application of Atangana-Baleanu FDEs. El-Ajou, Hadid, Al-Smadi et al. [19] developed approximated technique for solutions of population dynamics of Atangana-Baleanu fractional order.
Reproducing kernel Hilbert space (RKHS) is ideal for function approximation and estimate of fractional derivatives. In recent years, reproducing kernel functions (RKF) theory have been employed to solve linear and nonlinear fractional order problems, singularly perturbed problems, singular integral equations, fuzzy differential equations, and so on (see, e.g. [10,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]). However, there exists little discussion on numerical schemes for solving variable order Atangana-Baleanu FDEs.
In this paper, by using polynomials RKF, we will present a new collocation method for solving variable order Atangana-Baleanu FDEs.
This work is organized as follows. We summarize fractional derivatives and RKHS theory in Section 2. In Section 3, we develop RKF based collocation technique for Atangana-Baleanu variable order FDEs. Numerical experiments are provided in Section 4. Concluding remarks are included in the last section.
2.
Preliminaries to fractional derivatives and RKHS theory
Definition 2.1. Let H be a Hilbert function space defined on E. The function K:E×E→R is known as an RKF of space H if
If there exists a RKF in a Hilbert space, then the space is a RKHS.
Definition 2.2. Symmetric function K:E×E→R is known as a positive definite kernel (PDK) if n∑i,j=1cicjK(xi,xj)≥0 for any n∈N, x1,x2,…,xn∈E,c1,c2,…,cn∈R.
Theorem 2.1. [36] The RKF of an RKHS is positive definite. Besides, every PDK can define a unique RKHS, of which it is the RKF.
Definition 2.3. Let q>0. The one parameter Mittag-Leffler function of order q is defined by
Definition 2.4. Let q1,q2>0. The two-parameter Mittag-Leffler function is defined by
For the domains of convergence of the Mittag-Leffler functions, please refer to the following theorem.
Theorem 2.2. [37] For q1,q2>0, the two-parameter Mittag-Leffler function Eq1,q2(z) is convergent for all z∈C.
Definition 2.5. The Sobolev space H1(0,T) is defined as follows
Definition 2.6. The α∈(0,1) order Atangana- Baleanu fractional derivative of a function u∈H1(a,b) is defined
where M(α) is the normalization term satisfying M(0)=M(1)=1.
Theorem 2.3. [38] The function k(x,y)=(xy+c)m for c>0,m∈N is a PDK.
According to Theorem 2.1, there exists an associated RKHS Qm with k as an RKF.
3.
Collocation method
To solve (1.1), we will construct the RKF which satisfies the homogenous boundary condition.
Definition 3.1.
Theorem 3.1. The space Qm,0 is an RKHS and its RKF is expressed by
Proof. If Byk(x,y)=0 or Bxk(x,y)=0, then
If Byk(x,y)≠0, then
and naturally K(x,y)∈Qm,0.
For all u(y)∈Qm,0, we have u(y)∈Qm and Byu(y)=0.
We have
Thus, K(x,y) is the RKF of space Qm,0 and the proof is complete.
Suppose that L:Qm,0→H1 is a bounded linear operator. It is easy to proved that its inverse operator L−1 is also bounded since both Qm,0 and H1 are Banach spaces.
Choose N distinct scattered points in [0,1], such as {x1,x2,…,xN}. Put ψi(x)=K(x,xi),i=1,2,…,N. By using RKF basis, the RKF collocation solution uN(x) for (1.1) can be written as follows
where {ci}Ni=1 are undetermined constants.
Collocating (1.1) at N nodes x1,x2,…,xN provides N equations:
System (3.3) of equations is simplified to the matrix form:
where Aik=Lxψk(x)|x=xi,i,k=1,2,…,N, f=(f(x1,uN(x1)),f(x2,uN(x2)),…,f(xN,uN(xN)).
Theorem 3.2. If γ>0, then
and therefore matrix A can be computed exactly.
Proof. It is noticed that
Since RKF K(x,y) is a polynomials, matrix A in (3.3) can be calculated exactly. The proof is complete.
If f(x,u) is linear, then (3.3) is a system of linear equations and it is convenient to determine the value of the unknowns {ci}Ni=1. If f(x,u) is nonlinear, then (3.3) is a system of nonlinear equations, we solve it by using the tool "FindRoot" in soft Mathematica 11.0.
The residual function is defined as
Theorem 3.3. If a(x) and f(x,u)∈C4[0,1], then
where c>0 is a real number, h=max1≤i≤N∣xi+1−xi∣.
Proof. For the proof, please refer to [22].
4.
Numerical experiments
Three experiments are illustrated in this section to show the applicability and effectiveness of the mentioned approach. We take M(α)=1 in the following experiments.
Problem 4.1
Solve fractional linear initial value problems (IVPs) as follows:
where α(x)=0.5x+0.1, f(x)=ex(x2+x3+1)+M(α(x))1−α(x)2x2Eα(x),3(−α(x)1−α(x)xα(x))++M(α(x))1−α(x)6x3Eα(x),4(−α(x)1−α(x)xα(x)). The true solution of this equation is u(x)=x2+x3+1.
Selecting m=8,N=8, xi=iN,i=1,2,…,N, we apply our new method to Problem 4.1. The obtained numerical results are shown in Tables 1. The Mathematica codes for Problem 4.1 is provided as follows:
Problem 4.2
Solve the variable order fractional linear terminal value problems
where α(x)=sinx, f(x)=2(x4+2)+M(α(x))1−α(x)24x4Eα(x),5(−α(x)1−α(x)xα(x)). The exact solution is u(x)=x4+2.
Selecting m=8,N=8, xi=i−1N,i=1,2,…,N, the obtained absolute and relative errors of numerical results using our method are listed in Tables 2.
Problem 4.3
We apply our method to the nonlinear variable order fractional IVPs as follows
where α(x)=0.5x+0.1, f(x)=sinhx(x+x3+1)+M(α(x))1−α(x)xEα(x),2(−α(x)1−α(x)xα(x))+M(α(x))1−α(x)6x3Eα(x),4(−α(x)1−α(x)xα(x)). Its true solution is u(x)=x+x3+1.
Choosing m=8,N=8, xi=iN,i=1,2,…,N, we plot the absolute and relative errors in Figure 1.
5.
Conclusions
In this work, a new RKF based collocation technique is developed for Atangana-Baleanu variable order fractional problems. The proposed scheme is meshless and therefore it does not require any background meshes. From the numerical results, it is found that the accuracy of obtained approximate solutions is high and can reach to O(10−10). Also, for nonlinear fractional problems, our method can yield highly accurate numerical solutions. Hence, our new method is very effective and easy to implement for the considered problems.
Acknowledgments
The work was supported by the National Natural Science Foundation of China (No.11801044, No.11326237).
Conflict of interest
All authors declare no conflicts of interest in this paper.