Approximation Solution of Fractional Partial Differential Equations by Neural Networks

Abstract

Neural networks with radial basis functions method are used to solve a class of initial boundary value of fractional partial differential equations with variable coefficients on a finite domain. It takes the case where a left-handed or right-handed fractional spatial derivative may be present in the partial differential equations. Convergence of this method will be discussed in the paper. A numerical example using neural networks RBF method for a two-sided fractional PDE also will be presented and compared with other methods.

1. Introduction

In this paper, I will use neural network method to solve the fractional partial differential equation (FPDE) of the form:𝜕𝑢(𝑥,𝑡)𝜕𝑡=𝑐₊()𝜕𝑥,𝑡^𝛼𝑢(𝑥,𝑡)𝜕₊𝑥^𝛼+𝑐₋()𝜕𝑥,𝑡^𝛼𝑢(𝑥,𝑡)𝜕₋𝑥^𝛼+𝑠(𝑥,𝑡). (1.1)

On a finite domain 𝐿<𝑥<𝑅, 0≤𝑡≤𝑇. Here, I consider the case 1≤𝛼≤2, where the parameter 𝛼 is the fractional order (fractor) of the spatial derivative. The function 𝑠(𝑥,𝑡) is source/sink term [1]. The functions 𝑐₊(𝑥,𝑡)≥0 and 𝑐₋(𝑥,𝑡)≥0 may be interpreted as transport-related coefficients. We also assume an initial condition 𝑢(𝑥,𝑡=0)=𝐹(𝑥) for 𝐿<𝑥<𝑅 and zero Dirichlet boundary conditions. For the case 1<𝛼≤2, the addition of classical advective term −𝑣(𝑥,𝑡)𝜕𝑢(𝑥,𝑡)/𝜕𝑥 on the right-hand side of (1.1) does not impact the analysis performed in this paper and has been omitted to simplify the notation.

The left-hand (+) and right-hand (−) fractional derivatives in (1.1) are the Riemann-Liouville fractional derivatives of order 𝛼 [5] defined by𝐷^𝛼_𝐿+𝑓(𝑥)=𝑑^𝛼𝑓(𝑥)𝑑₊𝑥^𝛼=1Γ()𝑑𝑛−𝛼𝑑𝑥^𝑛𝑛𝑥_𝐿𝑓(𝜉)(𝑥−𝜉)^{𝛼+1−𝑛}𝐷𝑑𝜉,_𝛼𝑅−𝑓(𝑥)=𝑑^𝛼𝑓(𝑥)𝑑₋𝑥^𝛼=(−1)^𝑛Γ()𝑑𝑛−𝛼𝑑𝑥^𝑛𝑛𝑅_𝑥𝑓(𝜉)(𝑥−𝜉)^{𝛼+1−𝑛}𝑑𝜉, (1.2) where 𝑛 is an integer such that 𝑛−1≤𝛼≤𝑛. If 𝛼=𝑚 is an integer, then the above definitions give the standard integer derivatives, that is𝐷^𝑚_𝐿+𝑓(𝑥)=𝑑^𝑚𝑓(𝑥)𝑑𝑥^𝑚,𝐷^𝑚_𝑅−𝑓(𝑥)=(−1)^𝑚𝑑^𝑚𝑓(𝑥)𝑑𝑥^𝑚=𝑑^𝑚𝑓(𝑥)𝑑(−𝑥)^𝑚. (1.3)

When 𝛼=2, and setting 𝑐(𝑥,𝑡)=𝑐₊(𝑥,𝑡)+𝑐₋(𝑥,𝑡), (1.1) becomes the following classical parabolic PDE:𝜕𝑢(𝑥,𝑡)()𝜕𝜕𝑡=𝑐𝑥,𝑡²𝑢(𝑥,𝑡)𝜕𝑥²+𝑠(𝑥,𝑡). (1.4)

Similarly, when 𝛼=1 and setting 𝑐(𝑥,𝑡)=𝑐₊(𝑥,𝑡)−𝑐₋(𝑥,𝑡), (1.1) reduces to the following classical hyperbolic PDE:𝜕𝑢(𝑥,𝑡)()()𝜕𝑡=𝑐𝑥,𝑡𝜕𝑢𝑥,𝑡().𝜕𝑥+𝑠𝑥,𝑡 (1.5)

The case 1<𝛼<2 represents a superdiffusive process, where particles diffuse faster than the classical model (1.4) predicts. For some applications to physics and hydrology, see [2–4].

I also note that the left-handed fractional derivative of 𝑓(𝑥) at a point 𝑥 depends on all function values to the left of the point 𝑥, that is, this derivative is a weighted average of such function values. Similarly, the right-handed fractional derivative of 𝑓(𝑥) at a point 𝑥 depends on all function values to the right of this point. In general, left-handed and right-handed derivatives are not equal unless 𝛼 is an even integer, in which case these derivatives become localized and equal. When 𝛼 is an odd integer, these derivatives become localized and opposite in sign. For more details on fractional derivative concepts and definitions, see [1, 3, 5, 6]. Reference [7] provides a more detailed treatment of the right-handed fractional derivatives as well as a substantial treatment of the left-handed fractional derivatives.

Published papers on the numerical solution of fractional partial differential equation are scarce. A different method for solving the fractional partial differential equation (1.1) is pursued in the recent paper [4]. They transform this partial differential equation into a system of ordinary differential equations (method of lines), which is then solved using backward differentiation formulas. Another very recent paper, [8] develops a finite element method for two-point boundary value problem, and [1], finds the numerical solution of (1.1) by finite differences method.

2. Multilayer Neural Networks

The Rumelhart-Hinton-William’s multilayer network [9], that we consider here, is a feed-forward type network with connections between adjoining layers only. Networks generally have hidden layers between the input and output layers. Each layer consists of computational units. The input-output relationship of each unit is represented by inputs 𝑥_𝑖 output 𝑦, connection weights 𝑤_𝑖, threshold 𝜃, and differential function 𝜙 as follows:𝑦𝑥₁,…,𝑥_𝑛=^𝑚_𝑗=1𝑣_𝑗𝜙𝑊^𝑗−𝑋+𝜃_𝑗,where𝑥_𝑖∈𝑅,𝑊^𝑗∈𝑅^𝑛,𝑣_𝑗,𝜃_𝑗∈𝑅 (2.1) the learning rule of this network is known as the backpropagation algorithm [9], which is an algorithm that uses a gradient descent method to modify weight and thresholds as the error between the desired output and the output signal of the network is minimized. I generally use a bounded and monotonic increasing differentiable function which is called a radial basis function for each unit output function.

If a multilayer network has 𝑛 input units and 𝑚 output units, then the input-output relationship defines a continuous mapping from 𝑛-dimensional Euclidean space to 𝑚-dimensional Euclidean space. I call this mapping the input-output mapping of the network. I study the problem of network capability from the point of view of input-output mapping. It observed that for the study of mappings defined by multilayer networks, it is sufficient to consider networks whose 𝜙(𝑥) and whose output functions for input and output layers are linear.

3. Approximate Realization of Continuous Mappings by Neural Networks

Reference [10] considers the possibility of representing continuous mappings by neural networks whose output functions in hidden layers are sigmoid function, for example, 𝜙(𝑥)=1/(1+𝑒^−𝑥). It is simply noted here that general continuous mappings cannot be exactly represented by Rumelhart-Hinton-William’s networks. For example, if a real analytic output function such as the sigmoid function 𝜙(𝑥)=1/(1+𝑒^−𝑥) is used, then an input-output mapping of this network is analytic and generally cannot represent all continuous mappings.

Let points of 𝑛-dimensional Euclidean space 𝑅^𝑛 be denoted by 𝐱=(𝑥₁,…,𝑥_𝑛) and the norm of 𝐱 defined by ‖𝑥‖=(∑^𝑛_𝑖=0𝑥²_𝑖).

Definition 3.1 (see [11]). Let 𝐻=𝑅^𝑛 is a linear space. A function 𝑓∶𝐻→𝑅 is called a radial basis function that can be represented in the form 𝑓=𝑔𝑜𝜙, where 𝜙∶𝑅→𝑅, 𝑔∶𝑅^𝑛→𝑅, and 𝑔(𝑥)=‖𝑥−𝑤^𝑖‖², 𝑥,𝑤^𝑖∈𝑅^𝑛.
In this paper, I study approximate 𝑈 by using 𝑦 which is a representation model of neural networks, through analyzing previous theoretical studies. Also we present a study of convergent of solution 𝑈 by approximate solution 𝑦, where 𝜙(𝑤^𝑗,𝑥,𝜃) is radial basis function.
We also give some theorems for soliciting the conditions to converge the approximation solution for (1.1) by neural networks method.

Theorem 3.2 (Irie-Miyake, [6]). Let 𝜓(𝑥)∈𝐿¹(𝑅), that is, let 𝜓(𝑥) be absolutely integrable and 𝑢(𝑥₁,…,𝑥_𝑛)∈𝐿²(𝑅^𝑛). Let Ψ(𝜉) be integrable and 𝑈(𝑥₁,…,𝑥_𝑛) be Fourier transforms of 𝜓(𝑥) and 𝑢(𝑥₁,…,𝑥_𝑛), respectively. If  Ψ(1)≠0, then 𝑢𝑥₁,…,𝑥_𝑛=^∞_−∞⋯^∞_−∞𝜓^𝑛_𝑖=1𝑥_𝑖𝑤_𝑖−𝑤₀1(2𝜋)^𝑛Ψ(1)𝑈𝑤₁,…,𝑤_𝑛×exp𝑖𝑤₀𝑑𝑤₀𝑑𝑤₁⋯𝑑𝑤_𝑛. (3.1)

Theorem 3.3. Let 𝜙(𝑥) be a radial basis function, nonconstant, bounded, and monotone increasing continuous function. Let 𝐾 be a compact subset of 𝑅^𝑛 and 𝑢(𝑥₁,…,𝑥_𝑛) a real value continuous function on 𝐾. Then for an arbitrary 𝜀>0, there exists an integral 𝑁 and real constants 𝑣_𝑗,𝜃_𝑗,𝑤_𝑖𝑗(𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚) such that 𝑦(𝑥₁,…,𝑥_𝑛)=∑^𝑚_𝑗=1𝑣_𝑗𝜙[(∑^𝑛_𝑖=1(𝑤_𝑖𝑗−𝑥_𝑖)²]^1/2+𝜃_𝑗) satisfies max_𝑥∈𝐾|𝑦(𝑥₁,…,𝑥_𝑛)−𝑢(𝑥₁,…,𝑥_𝑛)|<𝜀. In other words, for an arbitrary 𝜀>0, there exists a three-layer network whose output function for the hidden layer is 𝜙(𝑥), whose output functions for input and output layers are linear, and which has an input-output function 𝑦(𝑥₁,…,𝑥_𝑛) such that max_𝑥∈𝐾|𝑦(𝑥₁,…,𝑥_𝑛)−𝑢(𝑥₁,…,𝑥_𝑛)|<𝜀.

Proof. First
Since 𝑢(𝐱)(𝐱=(𝑥₁,…,𝑥_𝑛)) is a continuous function on a compact subset 𝐾 of 𝑅^𝑛, 𝑢(𝑥) can be extended to be a continuous function on 𝑅^𝑛 with compact support.
If I operate the mollifier 𝜌_𝛼^∗ on 𝑢(𝑥), 𝜌_𝛼^∗𝑢(𝑥)is𝐶^∞-function with compact support. Furthermore, 𝜌_𝛼^∗𝑢(𝑥)→𝑢(𝑥)(𝛼→+0) uniformly on 𝑅^𝑛. Therefore, we may suppose 𝑢(𝐱) is a 𝐶^∞-function with compact support for proving Theorem 3.3. By the Paley-Wiener theorem [10], the Fourier transform 𝑈(𝐰)(𝐰=𝑤₁,…,𝑤_𝑛) of 𝑢(𝐱) is real analytic and, for any integer 𝑁, there exists a constant 𝐶_𝑁 such that |𝑈(𝑤)|≤𝐶_𝑁(1+|𝑤|)^−𝑁. (3.2) In particular 𝑈(𝑤)∈𝐿¹∩𝐿²(𝑅^𝑛).
I define 𝐼_𝐴(𝑥₁,…,𝑥_𝑛), 𝐼_∞,𝐴(𝑥₁,…,𝑥_𝑛), and 𝐽_𝐴(𝑥₁,…,𝑥_𝑛) as follows: 𝐼_𝐴𝑥₁,…,𝑥_𝑛=^𝐴_−𝐴⋯^𝐴_−𝐴𝜓^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2−𝑤₀1(2𝜋)^𝑛Ψ(1)𝑈𝑤₁,…,𝑤_𝑛×exp𝑖𝑤₀𝑑𝑤₀𝑑𝑤₁…𝑑𝑤_𝑛,𝐼_∞,𝐴𝑥₁,…,𝑥_𝑛=^𝐴_−𝐴⋯^𝐴_−𝐴^∞_−∞𝜓^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2−𝑤₀1(2𝜋)^𝑛Ψ(1)𝑈𝑤₁,…,𝑤_𝑛×exp𝑖𝑤₀𝑑𝑤₀𝑑𝑤₁⋯𝑑𝑤_𝑛,𝐽_𝐴𝑥₁,…,𝑥_𝑛=1(2𝜋)^𝑛𝐴_−𝐴^𝐴_−𝐴𝑈𝑤₁,…,𝑤_𝑛exp𝑖^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2𝑑𝑤₁⋯𝑑𝑤_𝑛, (3.3) where 𝜓(𝑥)∈𝐿¹ is defined by 𝜓(𝑥)=𝜙(𝑥/𝛿+𝛼)−𝜙(𝑥/𝛿−𝛼), 𝛿,𝛼>0.
The essential part of the proof of Irie-Miyake’s integral formula [1] is the equality 𝐼_∞,𝐴(𝑥₁,…,𝑥_𝑛)=𝐽_𝐴(𝑥₁,…,𝑥_𝑛), and this is derived from ^𝛼_−𝛼𝜓^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2−𝑤₀exp𝑖𝑤₀𝑑𝑤₀=exp𝑖^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2Ψ(1), (3.4) in our discussion, using the estimate of 𝑈(𝐰), it is easy to prove that lim_𝐴→∞𝐽_𝐴(𝑥₁,…,𝑥_𝑛)=𝑢(𝑥₁,…,𝑥_𝑛) uniformly on 𝑅^𝑛. Therefore, lim_𝐴→∞𝐼_∞,𝐴(𝑥₁,…,𝑥_𝑛)=𝑢(𝑥₁,…,𝑥_𝑛) uniformly on 𝑅^𝑛. I can state that for any 𝜀>0, there exists 𝐴>0 such that max_{𝑥∈𝑅𝑛}𝐼_∞,𝐴𝑥₁,…,𝑥_𝑛−𝑢𝑥₁,…,𝑥_𝑛<𝜀2. (3.5)
Second
I will approximate 𝐼_∞,𝐴 by finite integrals on 𝐾. For 𝜀>0, fix 𝐴 which satisfies (3.5). For 𝐴^′>0, set 𝐼_𝐴′,𝐴𝑥₁,…,𝑥_𝑛=^𝐴_−𝐴⋯^𝐴_−𝐴^𝐴′_−𝐴′𝜓^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2−𝑤₀×1(2𝜋)^𝑛Ψ(1)𝑈𝑤₁,…,𝑤_𝑛×exp𝑖𝑤₀𝑑𝑤₀𝑑𝑤₁⋯𝑑𝑤_𝑛. (3.6) I will show that, for 𝜀>0, I can take 𝐴^′>0 so that max_𝑥∈𝐾𝐼_𝐴′,𝐴𝑥₁,…,𝑥_𝑛−𝐼_∞,𝐴𝑥₁,…,𝑥_𝑛<𝜀2. (3.7) Using the following equation: ^𝐴′_−𝐴′𝜓^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2−𝑤₀exp𝑖𝑤₀𝑑𝑤₀=^{∑𝑛𝑖=1𝑥𝑖𝑤𝑖+𝐴′}_{∑𝑛𝑖=1𝑥𝑖𝑤𝑖−𝐴′}𝜓(𝑡)exp(−𝑖𝑡)𝑑𝑡⋅exp𝑖^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2, (3.8) the fact 𝑈(𝑥)∈𝐿¹ and compactness of [−𝐴,𝐴]^𝑛×𝐾, we can take 𝐴^′ so that ^𝐴′_−𝐴′𝜓^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2−𝑤₀exp𝑖𝑤₀𝑑𝑤₀−^∞_−∞𝜓^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2−𝑤₀exp𝑖𝑤₀𝑑𝑤₀≤𝜀(2𝜋)²|Ψ(1)|2∫^∞_−∞⋯∫^∞_−∞|𝑈(𝑥×)|𝑑𝑥+1^𝐴−𝐴⋯^𝐴_−𝐴|𝑈(𝑥)|𝑑𝑥on𝐾. (3.9) Therefore, max_𝑥∈𝐾𝐼_𝐴′,𝐴𝑥₁,…,𝑥_𝑛−𝐼_∞,𝐴𝑥₁,…,𝑥_𝑛≤𝜀2∫^∞_−∞⋯∫^∞_−∞|𝑈(𝑥×)|𝑑𝑥+1^𝐴−𝐴⋯^𝐴_−𝐴|𝑈(𝑥𝜀)|𝑑𝑥<2. (3.10)
Third
From (3.5) and (3.7), I can say that for any 𝜀>0, there exists 𝐴,𝐴^′>0 such that max_𝑥∈𝐾𝑢𝑥₁,…,𝑥_𝑛−𝐼_𝐴′,𝐴𝑥₁,…,𝑥_𝑛<𝜀, (3.11)𝑢(𝐱) can be approximated by the finite integral 𝐼_𝐴′,𝐴(𝑋) uniformly on 𝐾. The integral of 𝐼_𝐴′,𝐴(𝑋) can be replaced by the real part and is continuous on [−𝐴^′,𝐴^′]×⋯×[−𝐴,𝐴]×𝐾, hence 𝐼_𝐴′,𝐴(𝑋) can be approximated by the Riemann sum uniformly on 𝐾. Since 𝜓^𝑛_𝑖=1𝑥_𝑖−𝑤_𝑖^21/2−𝑤₀∑=𝜙^𝑛𝑖=1𝑥_𝑖−𝑤_𝑖^21/2𝛿−𝑤₀∑+𝛼−𝜙^𝑛𝑖=1𝑥_𝑖−𝑤_𝑖^21/2𝛿−𝑤₀−𝛼, (3.12) the Riemann sum can be represented by a three-layer network. Therefore 𝑢(𝐱) can be represented approximately by three-layer network.

Theorem 3.4. Let 𝐻 be a linear space. The set of radial basis functions denoted by Φ={exp𝑜𝜙∶𝜙∈𝐻^∗} is fundamental in 𝐶(𝐻).

Proof. Let 𝑇 denotes the linear span of the set, Φ as I will prove 𝑇 an algebra in 𝐶(𝐻), for 𝑥∈𝐻 and 𝜙,𝜓∈𝐻^∗ (where 𝐻^∗ is the dual space of 𝐻) [11] 𝑥)=(𝑥)⋅(𝑥)𝜙(𝑥)𝜓(𝑥)𝜙(𝑥)(𝑥)𝑥).exp𝑜𝜙exp𝑜𝜓(exp𝑜𝜙exp𝑜𝜓=exp⋅exp=exp+𝜓=exp𝜙+𝜓( (3.13) Let 𝜙₀ be the zero functional in 𝐻^∗, then (exp𝑜𝜙₀)=1 for all 𝑥∈𝐻. Hence, 𝑇 contains constants.
Now if 𝑥,𝑦∈𝐻 and 𝑥≠𝑦, then there exists 𝜙∈𝐻^∗ such that 𝜙(𝑥−𝑦)≠0. 𝜙(0)𝜙(𝑥)𝑦𝜙(𝑥𝑦𝜙(𝑥exp𝑥−𝑦≠exp=1,exp−𝜙≠1,exp)⋅exp−𝜙≠1,exp)𝜙𝑦(𝑥)𝜙𝑦,exp≠1,exp𝜙≠exp (3.14) hence 𝑇 separates point of 𝐻.
Now, I will prove each basic neighborhood of a fixed point 𝑢 in 𝐶(𝐻) intersects 𝑇; 𝐾 is compact set such that 𝐾⊂𝐻, a basic neighborhood of 𝑢 corresponding to 𝐾, and has the form 𝐵_𝐾,𝜀={𝑔∈𝐶(𝐻)∶‖𝑢−𝑔‖_𝐾<𝜀, now restrict 𝑢 and all members of 𝑇 to 𝐾, then {ℎ/𝐾∶ℎ∈𝑇} is still on algebra-containing constants separating of 𝐾, hence {ℎ/𝐾∶ℎ∈𝑇} is dense in 𝐶(𝐾), and consequently, 𝑇 intersects 𝐵_𝐾,𝜀, as required. Therefore, 𝑇 is fundamental in 𝐶(𝐻) and required.

4. Numerical Methods

Conceding a feed forward network with input layer of a single hidden layer, and an output layer consisting of a single unit, I have purposely chosen a single output unit to simplify the exposition without loss of generality.

The network is designed to perform a nonlinear mapping form the input space to the hidden space, followed by a linear mapping from the hidden space to the output space.

Given a set of 𝑁 different points {𝑥_𝑖∈𝑅^𝑛,𝑖=1,2,…,𝑛} and a corresponding set 𝑁 real numbers {𝑢_𝑖∈𝑅^𝑛,𝑖=1,2,…,𝑛}, we find a function 𝑓: 𝑅^𝑛→𝑅 that satisfied the interpolation conditions𝑓𝑥_𝑖=𝑦_𝑖,𝑖=1,2,…,𝑛. (4.1)

For strict interpolation as specified here, the interpolating surface (i.e.; function 𝑓(𝑥)) is constraining to pass through all the training data points, initial 𝑤⁽⁰⁾∈𝑅^𝑛+𝑚, 𝜃⁽⁰⁾,𝑑⁽⁰⁾∈𝑅, 𝑣⁽⁰⁾∈𝑅^𝑛, we will use algorithm of backpropagation neural network [11] to find the best weights 𝑊,𝑉,𝑑, and 𝜃, inserting the interpolation conditions of (1.1) in (2.1), we obtain the following set of simultaneous linear equations for the unknown coefficients of the expansion𝜙₁₁𝜙₁₂…𝜙_1𝑛𝜙₂₁𝜙₂₂…𝜙_2𝑛𝜙…………_𝑛1𝜙_𝑛2𝜙_𝑛𝑛𝜈₁𝜈₂…𝜈_𝑛=𝑦₁𝑦₂…𝑦_𝑛, (4.2) where 𝜙_𝑖𝑗=𝜙(‖𝑊^𝑗_𝑖−𝑋_𝑖‖), (𝑖,𝑗)=1,2,…,𝑛.

Compute the error 𝐸=(1/2)∑^𝑁_𝑖=1(𝑦_𝑖−𝑢_𝑖)², and 𝛿_𝑖=(𝑦_𝑖−𝑢_𝑖)^∗𝜙^′(𝑦_𝑖).

Calculate Δ𝜈_𝑖=𝛼𝛿_𝑖^∗𝑍_𝑖=𝛼𝛿_𝑖^∗‖𝑋_𝑖−𝑊_𝑖𝑗‖(𝛼is training rat 0<𝛼<1),Δ𝑑_𝑗=𝛼^∗𝛿_𝑗,𝜌_𝑖=^𝑁_𝑗=1𝛿_𝑗𝑉_𝑗,𝛽_𝑖=𝜌_𝑖𝛼^′𝑋_𝑖−𝑊_𝑖𝑗,Δ𝑊_𝑖𝑗=𝛼𝛽_𝑖𝑥_𝑖,Δ𝜃_𝑖=𝛼𝛽_𝑖,𝑉^new_𝑖=𝑉^old_𝑗+Δ𝑉_𝑗,𝑊^new_𝑖𝑗=𝑊^old_𝑖𝑗+Δ𝑊_𝑖𝑗,𝜃^new_𝑖=𝜃^old_𝑗+Δ𝜃_𝑖,𝑑^new_𝑖=𝑑^old_𝑗+Δ𝑑_𝑖. (4.3)

After that the best weights 𝑊, 𝑉, 𝜃, 𝑑 are fined. Given the substitute in (2.1), the approximate solution is found.

The above method is supposed to work in this situation. In such case, the values of absence of the exact solution of independent variables will be substituted in the boundary conditions in order to get the exact values. Those values can be used in the phase of training of the considered backpropagation neural network. The approach is proved to get good weights of (2.1). The radial basis function in Theorems 3.3 and 3.4 was used in order to obtain converge training of the backpropagation neural network.

5. Numerical Example

The following two-sided fractional partial differential equation𝜕𝑢(𝑥,𝑡)𝜕𝑢=𝑐₊()𝜕𝑥,𝑡^1.8𝑢(𝑥,𝑡)𝜕₊𝑥^1.8+𝑐₋()𝜕𝑥,𝑡^1.8𝑢(𝑥,𝑡)𝜕₋𝑥^1.8+𝑠(𝑥,𝑡) (5.1) were considered on a finite domain 0<𝑥<2 and 𝑡>0 with the coefficient functions 𝑐₊(𝑥,𝑡)=Γ(1.2)𝑥^1.8 and 𝑐₋(𝑥,𝑡)=Γ(2−𝑥)𝑥^1.8, and the forcing function 𝑠(𝑥,𝑡)=−32𝑒^−𝑡𝑥²+(2−𝑥)²−2.5𝑥³+(2−𝑥)³+25𝑥22⁴+(2−𝑥)⁴, (5.2) initial condition 𝑢(𝑥,0)=4𝑥²(2−𝑥)², and boundary conditions 𝑢(0,𝑡)=𝑢(2,𝑡)=0. This fractional PDE has the exact solution 𝑢(𝑥,𝑡)=4𝑒^−𝑡𝑥²(2−𝑥)², which can be verified applying the fractional formulas 𝐷^𝛼_𝐿+(𝑥−𝐿)^𝑝=Γ𝑝+1Γ()𝑝+1−𝛼𝑥−𝐿^𝑝−𝛼,𝐷^𝛼_𝑅−(𝑅−𝑥)^𝑝=Γ𝑝+1Γ()𝑝+1−𝛼𝑅−𝑥^𝑝−𝛼 (5.3) (see [1]).

Table 1 shows the training data by values of the boundary conditions. Table 2 shows the comparisons between exact solution 𝑢(𝑥_𝑖,𝑡_𝑖) and the approximated solution test points 𝑦(𝑥_𝑖,𝑦_𝑖) at Δ𝑥=0.2, Δ𝑡=0.1. Table 3 compares maximum error between approximate solution by artificial neural networks method and finite difference numerical method [1].

Table 1 

The training data by values of the boundary conditions.

Table 2 

The comparisons between exact solution 𝑢(𝑥_𝑖,𝑡_𝑖) and the approximated solution test points 𝑦(𝑥_𝑖,𝑦_𝑖) at Δ𝑥_𝑖=0.2, Δ𝑡_𝑖=0.1.

Table 3 

Compares maximum error between approximate solutions.

6. Conclusion

Referring back to Tables 1–3, it is clear that the approximation of fractional partial FPDE using neural network with RBF obtains a good approximated solution. The converge of this method is given as well. In addition, the discussed method is able to solve fractional partial differential equations with more than two variables where many other methods fail, clearly. The suggested method provides a general approximated solution to the interval domain and depends on boundary conditions.