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Application of ANNs approach for wave-like and heat-like equations

  • Ahmad Jafarian EMAIL logo and Dumitru Baleanu
Published/Copyright: December 29, 2017
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Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

Artificial neural networks are data processing systems which originate from human brain tissue studies. The remarkable abilities of these networks help us to derive desired results from complicated raw data. In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems. It should be noted that many physical phenomena such as coupling currents in a flat multi-strand two-layer super conducting cable, non-homogeneous elastic waves in soils and earthquake stresses, are described by initial-boundary value wave and heat partial differential equations with variable coefficients. To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned initial-boundary value problem. Finally, several computer simulations confirmed the theoretical results and demonstrating applicability of the method.

Keywords: wave-like and heat-like equations; bivariate power series polynomial; artificial neural network; criterion function; back-propagation learning algorithm
MSC 2010: 07.05.Mh; 02.60.Jh

1 Introduction

As known, differential equations occur in many science phenomena. Whenever, there is a meaningful relation between different values, states or times in which the rate of variable changes at different times or states are known, it can be modeled via differential equations. Hence, various applications of differential equations have led mathematicians and engineer scientist to focus on appropriate methods in solving these kinds of mathematical problems. More complicated differential equations arising from the modeling of complex phenomena can not be solved simply by existing conventional methods. Therefore, finding alternative numerical techniques seems to be necessary. During recent decades, various numerical techniques have been used for approximating different types of differential equations in linear and nonlinear cases. Among these, we can mention the homotopy perturbation method [2, 4, 5], variational iteration method [21, 22, 26], Adomian decomposition method [1, 3, 10], etc. An effective method in solving a wide variety of complicated mathematical problems is artificial neural networks (ANNs) approach. Since, these networks have high efficiency in approximating solutions of mathematical problems. Recently, some structures of neural networks have been applied for solving variety of mathematical problems [11, 12, 13].

In this paper, a suitable structure of neural networks will be applied for solving initial-boundary one-dimensional wave-like and heat-like equations. The heat-like equation can be a mathematical modeling for temperature changes of the composite materials. The solution of the mentioned partial differential equation is proportional to kinetic energy of particles in the material. Meanwhile, the wave-like equation can describe moving the particles under Hook’s law in the disordered systems. In this kind of equation, the standard initial and boundary conditions can be considered. In recent years, some researchers have solved numerically these problems [14, 18, 19, 20, 24, 25]. To do this, a four-layer feed-forward neural network corresponding to the mentioned bivariate power series solution is designed satisfying both initial and boundary conditions. The designed neural architecture can easily approximate solution functions on the space of unknown constant coefficients using a suitable learning algorithm. In other words, after discretizing the differential domain using a standard rule and doing some simplifications, the origin problem is transformed to solve a minimizing optimization problem. The yield nonlinear problem is solved iteratively using a back propagation learning rule, which is based on the gradient descent method. Approximating the constant series coefficients lead to finding the mentioned series solution on the given differential domain. In Section 2, we will first have an overview of artificial neural nets and their computational process. Then, having the introduction of the proposed neural architecture, the numerical solution of the mentioned differential equations will be considered using the defined combination iterative method. Two numerical examples with computer simulations are presented in Section 3. Also, to better express the effectiveness of the presented technique, the obtained numerical results will compare via the ones achieved from another classical method. Finally, conclusions and recommendation for future research are presented in Section 4.

2 Description of the method

The general purpose of this section is to introduce a combination iterative method for approximating solution of two well known partial differential equations. In order to better clarify all the fundamental mathematical features of the method presented in this research, we first deal with more general theories of neural networks issue. The proposed iterative technique which is based on the combination of power series method and a modification of ANNs approach, is used to find approximate solutions of boundary-initial value wave-like and heat-like equations on a given closed domain.

2.1 Basic idea of ANNs

Artificial neural networks theory revolves around the idea that certain key processing properties of human being brain can be modeled and applied to approximate computational methods using biological processes. In other words, artificial neural networks attempt to get knowledge of the relation between a set of data through training and finally store the knowledge gained for similar purpose. The main ideas of these networks are partly inspired by a way biological nervous system functions, to process data and information for learning and createing knowledge. In a neural network, simple processing elements are known as “neurons”, which can display complex global behavior determined by the connection between processing elements and element parameters. A neural net can not be adaptive itself. The practical application of this network is needed to enable employing algorithms which are designed to alter and adjust unknown parameters. For this purpose, using knowledge of computer programming, we can design a structure that acts as a nervous system. So, a learning algorithm is defined for networks by creating a network of interconnected artificial neurons. Mentioned networks have exhibited high performance on estimation and approximation. There are many references on neural nets field, see for example Ref. [6, 7, 8].

Now, let’s consider the neural architecture shown in Figure 1. This network is a four-layer feed-forward neural architecture with two input signals and one output neuron. Each input signal is multiplied by its respective weight values, and these weighted signals are then perfectly summed to produce the next layer’s inputs. In other words, each hidden or output layer receives its inputs from the previous weighted neurons and then presents it to a suitable activation function. The network output can be calculated. The remarkable notation in this area is that the bias term in output layer is added with the weighted signals of second hidden layer and then makes up the overall input of the final layer. According to the above, each neuron’s input-output relation can be written as follows:

Figure 1 The designed neural network architecture
Figure 1

The designed neural network architecture

  • Input units:

    o11=x,(1)
    o21=t.

  • First hidden units:

    o1,i2=(o11)i,i=1,...,n,(2)
    o2,j2=(o21)j,j=1,...,m.

  • Second hidden units:

    oi,j3=o1,i2.o2,j2,i=1,...,n,j=1,...,m.(3)

  • Output unit:

    N(x,t)=∑i=1n∑j=1m(ai,j.oi,j3)+∑i=1n(ai,0.o1,i2)+∑j=1m(a0,j.o2,j2)+a0,0.(4)

The designed neural structure is a prototype model, which should be created having some minor changes caused by conditions of problem. In other words, by presenting boundary or initial conditions of a given partial differential equation problem, the neural architecture will be ready to learn. This will cause the network output desirability to the solution of the mentioned math problem.

2.2 Implementation of the method

As mentioned before, the main idea of this study is to apply the designed multi-layer feed-forward neural architecture to the function approximation of two famous types of partial differential equations. Consider the one-dimensional wave-like Equation (5) and heat-like Equation (6):

∂u∂t2−k(x,t)∂2u∂x2=h(x,t),0<x<1,t>0,(5)
∂u∂t−k(x,t)∂2u∂x2=h(x,t),0<x<1,t>0,(6)

subject to the initial conditions:

u(x,0)=g0(x),∂u(x,t)∂t|t=0=g1(x),

and the boundary conditions:

u(0,t)=f0(t),u(1,t)=f1(t).

In the above relations, k and h are given continuous positive and real-valued functions, respectively. The defined wave-like and heat-like equations are more applicable in modeling different physical phenomena. The main problem occurs when we encounter more complicated modeled problems of these types, in which the existing analytical or numerical methods can not solve them. Therefore, finding an effective alternative method seems to be necessary. In the last decade, some modifications of the power series method have been employed for solving several types of partial differential equations with initial or boundary conditions [16, 17, 23]. Extensive studies on these polynomials, show that this series solution technique is really a powerful tool for solving complex mathematical problems. As previously mentioned, a combination of these polynomials and ANNs approach will be considered as an iterative method for solving the above partial differential equations. The basic idea in this issue is that the solution function u(x, t) on domain Ω = [0, 1]×[0, T], can be completely represented in a polynomial series of degree (n, m) as:

un,m(x,t)=∑i=0n∑j=0mai,jxitj,(7)

for the constant coefficients ai,j(for i = 0, …, n; j = 0, …, m). It should be noted that the solution function can be represented as the power series (7) if and only if it is complex differentiable in the open set (0, 1) × (0, T). It is reasonable to consider that the designed neural architecture are fully associated with the polynomial series (7). The interesting point in this approach is that the growing degree of basis polynomials, increases the accuracy of the recommended combination method. It must be considered that, this work will lead to more complex relations.

2.2.1 Discretization of the problems

Power series method is based on the fact that any modification of these series before being used, must be satisfied in the initial or boundary conditions. For the introduced partial differential equations, the trial solution is written as follows:

u~(x,t)=A(x,t)+x(1−x)t(N(x,t)−N(x,0)),(8)

where

A(x,t)=(1−x)f0(t)+xf1(t)+g0(x)−(1−x)g0(0)+xg0(1)+t{g1(x)−[(1−x)g1(0)+xg1(1)]},

in which the function A(x, t) has been chosen in a manner that satisfies in the both initial and boundary conditions, simultaneously. The introduced trial function involves the given feed-forward architecture that satisfies in both initial and boundary conditions. Supposedly u͂(x, t) symbolizes the trial solution with adjustable parameter ai,j, the problem is transformed from the original construction to an unconstrained one by direct substitution (8) in the primary equations. So, the Eqs. (5) and (6) are shape changes into the form (9) and (10), respectively:

u~tt(x,t)−k(x,t)u~xx(x,t)=h(x,t),(x,t)∈Ω,(9)
u~t(x,t)−k(x,t)u~xx(x,t)=h(x,t),(x,t)∈Ω,(10)

where

u~xx(x,t)=Axx(x,t)+t(−2(N(x,t)−N(x,0))+(2−4x)(Nx(x,t)−Nx(x,0))+(x−x2)(Nxx(x,t)−Nxx(x,0))),u~t(x,t)=At(x,t)+x(1−x){N(x,t)−N(x,0)+tNt(x,t)},u~tt(x,t)=Att(x,t)+x(1−x)(2Nt(x,t)+tNtt(x,t)),

and

Nt(x,t)=∑i=0n∑j=1mjai,jxitj−1,Ntt(x,t)=∑i=0n∑j=2mj(j−1)ai,jxitj−2,Nx(x,t)=∑i=1n∑j=0miai,jxi−1tj,Nxx(x,t)=∑i=2n∑j=0mi(i−1)ai,jxi−2tj,At(x,t)=(1−x)f0′(t)+xf1′(t)+g1(x)−((1−x)g1(0)+xg1(1)),Att(x,t)=(1−x)f0″(t)+xf1″(t),Axx(x,t)=g0″(x)+tg1″(x).

Now, we intend to define a set of acceptable mesh points for the discretization of Equations. (9) and (10). For positive integers n′ and m′, let Ωp, q be a partition of square Ω with the mesh points (xp, tq) = (pn′,qTm′), (for p = 0, …, n′; q = 0, …, m′). Substituting collocation point (xp, tq) into the resulted relations, reduces the problems into the following systems of equations:

u~tt(xp,tq)−k(xp,tq)u~xx(xp,tq)=h(xp,tq),p=0,...,n′,(11)
u~t(xp,tq)−k(xp,tq)u~xx(xp,tq)=h(xp,tq),q=0,...,m′.(12)

Continuing, we intend to construct an iterative procedure with the help of artificial neural networks approach to solve the resulting systems.

2.2.2 Proposed error function

As it is known, in iterative methods a valid criterion is required to measure the produced error of each iteration. Here, the differentiable least mean square (LMS) function is employed to measure the accuracy of solutions. This rule is stated for wave-like and heat-like equations, respectively as follows:

EWp,q=12u~tt(xp,tq)−k(xp,tq)u~xx(xp,tq)−h(xp,tq)2,(13)
EHp,q=12u~t(xp,tq)−k(xp,tq)u~xx(xp,tq)−h(xp,tq)2.(14)

Minimizing the defined error functions over the space of possible weight parameters can be an interested issue. To do this, a set of training rules is build to minimize EW and EH by adaptively adjusting the network parameters. Hence, one suitable error correction technique must be essentially employed to achieve this goal. More details concerning minimizing techniques can be found in Ref. [9].

2.2.3 Proposed learning algorithm

What makes this particular use of neural networks so attractive in many applications is that they express the ability to learn, though this remarkable property might be challenged by some researchers. In terms of ANNs, “learning” simply means changing the weights and biases of the network in response to some input data. Once a particular learning algorithm succeeds, programming the network to have a particular unequivocal performance is not vitally important. In other words, we need no prior knowledge to adjust the weights and biases. The designed neural architecture adjusts its parameters for a learning algorithm. That is, the error alters as the weights and bias term are changed. In this sense, the neural network learns from experience. Therefore, the back-propagation algorithm is the most widely-used method for feed-forward networks. The learning rules are mathematical formalizations that are believed to be more effective in the ANN’s performance. To fine-tune the neural network, the network parameters are first quantified with arbitrary initial guesses; then, the neural network calculates the output for each input signal. Next, the defined error rule is employed by substituting the proposed network model instead of the solution function in the origin problem. To train the present network, we have employed an optimization technique that in turn required the computation of the gradient of the error with respect to the net parameters.

Now, a suitable error correction rule must be initially used for single units training. This rule essentially drives the output error of the network to zero. We start with the classical generalized delta learning rule and give a brief description for its performance. Throughout this section, an attempt is made to point out the criterion function that is minimized by using this rule. Learning in neural nets is appropriate selecting the connection weights, which yields to minimize the error function on a set of mesh points. During the training, the initial parameters ai,j are put into the network and flow through the network generating a real value on the output unit. As seen in the last part, the calculated output is compared with the desired one, and an error is computed. The differentiable cost functions EW and EH are always decreasing in the opposite direction of its derivative. It means that if we want to find one of the local minima of this function starting from a initial guess. We employ the supervised back-propagation learning algorithm to reach this goal. The mentioned self learning mechanism starts with randomly quantifying the initial parameters ai,j (for i = 0, …, n; j = 0, …, m). The mentioned algorithm is well presented for wave-like equation as follows:

ai,j(r+1)=ai,j(r)+Δai,j(r),(15)
Δai,j(r)=−η.∂EWp,q∂ai,j+γ.Δai,j(r−1),p=0,...,n′;q=0,...,m′,

where η and γ are the learning rate and momentum term, respectively. In the above, the index r in ai,j(r) ascribes to the repetition number and the subscript i, j in ai,j is the label of the training connection weight. Moreover, ai,j(r + 1) and ai,j(r) depict the adjusted and current weight parameter, respectively. To complete the derivation of learning procedure for the output layer weights, the above partial derivative can be expressed as follows:

∂EWp,q∂ai,j=(u~tt(xp,tq)−k(xp,tq)u~xx(xp,tq)−h(xp,tq))×∂u~tt(xp,tq)∂ai,j−k(xp,tq)∂u~xx(xp,tq)∂ai,j,

where

∂u~tt(xp,tq)∂ai,j=∂Att(xp,tq)∂ai,j+x(1−x)2∂Nt(xp,tq)∂ai,j+t∂Ntt(xp,tq)∂ai,j,
∂u~xx(xp,tq)∂ai,j=∂Att(xp,tq)∂ai,j+tq−2∂N(xp,tq)∂ai,j−∂N(xp,0)∂ai,j+(2−4xp)∂Nx(xp,tq)∂ai,j−∂Nx(xp,0)∂ai,j+(xp−xp2)∂Nxx(xp,tq)∂ai,j−∂Nxx(xp,0)∂ai,j,

and

∂N(xp,tq)∂ai,j=xpitqj,∂Nx(xp,tq)∂ai,j=ixpi−1tqj,∂Nxx(xp,tq)∂ai,j=i(i−1)xpi−2tqj,∂Nt(xp,tq)∂ai,j=jxpitqj−1,∂Ntt(xp,tq)∂ai,j=j(j−1)xpitqj−2.

The above computational process can similarly be employed for heat-like equation. To prevent taking much of the time on this part, the behavior of adjusting weight parameters for this equation is not provided. It should be mentioned clearly that Matlab v7.10 is a high quality and easy to use mathematical computing software, which researchers and students can employ to omit wasting time and enhance the accuracy of calculations.

3 Numerical examples

In this section two test problems with computer simulations are provided to illustrate ability and accuracy of the iterative technique proposed in this research. Furthermore, the obtained numerical results of this technique will be compared with those obtained by Taylor series method [15]. Here, the mean absolute error Emidi.e.:

Emid=1(n′+1)(m′+1)∑i=0n′∑j=0m′|u(xi,tj)−u~(xi,tj)|,

will be implemented to compare the effectiveness of both methods.

Example 3.1

The following heat-like equation is considered:

∂u∂t−x26.∂2u∂x2=0,0<x<1,t>0,

with initial conditions:

ut(x,0)=u(x,0)=x3,

and the boundary conditions:

u(0,t)=0,u(1,t)=et.

Note that the exact solution of the problem is given as u(x, t) = x3et. Now, we intend to approximate the solution function by using the defined combination method on the domain Ω = [0, 1]2. Here, we use the regular discretization technique on the given differential domain in x and t directions. We intend to show that the proposed four-layer feed-forward neural architecture is sufficient to solve the defined math problem. Hence, the incremental learning process begins to work by quantifying the connection weights ai,j (for i = 0, …, n; j = 0, …, m) with small real-valued random constants. The convergence speed of back-propagation is directly related to the learning rate and momentum constant parameters. The optimal tuning parameters for fast convergence of back-propagation gradient descent search is the inverse of the largest eigenvalue of the Hessian matrix of the defined error function, evaluated at the local point. Thus, the norm of the converged weight vector gives a good estimate of learning rate in back-propagation. In this study, for better comparison of the obtained numerical and simulation results, we had to use same quantities for rate and momentum parameters. This work makes easy to compare the obtained results from using different initial parameters and iterations numbers. In other words, using same valued learning rate and momentum constant makes better comparison of the obtained results. Then, the training patterns were used to successively adjust the connection weights until a suitable solution was found. Typically, more than one step using the training set is needed to derive an appropriate solution. To demonstrate the accuracy of technique presented in the previous section, the indicated mean absolute errors for different network parameters are shown in Table 1, for n = m and n′ = m″. The absolute errors between the approximate and exact solutions on the mesh points are numerically compared for n = m = 5 and n′ = m′ = 10 in Table 2. As can be seen, with increasing number of iterations the ANNs approach offers more accurate approximations rather than the Taylor series method of order 5. The indicated error function is plotted in Figure 1, for m = n = 5 and m′ = n′ = 5. It can be easily concluded that by increasing the number of iterations, the network error is rapidly reduced until it go to zero. The exact and approximate solutions are plotted in Figure 3. The performance of proposed neural structure for different control elements is using the mean absolute error function in Figure 4.

Figure 2 The cost function for different iteration steps
Figure 2

The cost function for different iteration steps

Figure 3 Exact and approximate solutions for r = 1000
Figure 3

Exact and approximate solutions for r = 1000

Figure 4 Performance of proposed neural architecture over different neural elements for r = 1000
Figure 4

Performance of proposed neural architecture over different neural elements for r = 1000

Table 1

Measured Emid errors for different number of network elements

– – – – – –n = 5– – – – – –– – – – – –n = 7– – – – – –
rn′ = 5n′ = 10n′ = 15n′ = 5n′ = 10n′ = 15
10002.9323e – 082.7833e – 081.6822e – 083.5860e – 113.3621e – 112.7600e – 11
25008.7340e – 097.0041e – 096.9740e – 098.1071e – 126.0731e – 124.9617e – 12
50001.1375e – 095.6823e – 103.0116e – 103.2074e – 129.4698e – 133.6220e – 13
75007.9910e – 116.1845e – 116.4731e – 115.8193e – 132.1633e – 139.0790e – 14
100009.2155e – 115.0160e – 111.7147e – 116.0514e – 145.7311e – 144.9321e – 14
– – – – – –n = 9– – – – – –– – – – – –n = 11– – – – – –
rn′ = 5n′ = 10n′ = 15n′ = 5n′ = 10n′ = 15
10001.6340e – 121.4333e – 121.2173e – 125.2961e – 144.8245e – 143.6175e – 14
25007.1327e – 135.0108e – 132.9388e – 132.3799e – 149.3693e – 158.6722e – 15
50002.9110e – 131.2940e – 138.4830e – 141.0861e – 146.7127e – 154.2973e – 15
75007.6931e – 146.1577e – 145.7827e – 147.1911e – 155.0409e – 152.2500e – 15
100004.2780e – 143.0091e – 141.3786e – 145.7860e – 152.0617e – 159.8476e – 16

Table 2

Absolute error comparison for different number of iterations

ANNs approach
X = tTE method– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
r = 1000r = 2500r = 5000r = 10000
0.11.4090e – 123.5275e – 086.5374e – 097.3595e – 118.0373e – 12
0.27.3195e –103.2403e – 088.2796e – 093.4572e – 108.4933e – 12
0.32.8555e – 083.1375e – 088.7914e – 097.6941e –101.0855e – 11
0.43.8598e – 071.9725e – 089.0478e – 096.7459e – 102.4573e – 11
0.52.9193e – 061.7860e – 085.4376e – 096.1150e – 103.0391e – 11
0.61.5293e – 052.3485e – 086.7328e – 099.3688e – 105.9043e –11
0.76.2183e – 052.0389e – 086.1107e – 093.7592e – 107.6900e –11
0.82.1005e – 041.3481e – 087.6833e – 097.8927e – 116.8627e – 11
0.96.1590e – 042.1768e – 088.3284e – 098.9380e –112.0981e – 11

Example 3.2

Consider the following one-dimensional wave-like equation:

∂2u∂t2−x22.∂2u∂x2=0,0<x<1,t>0,

with initial conditions:

u(x,0)=x+x2,ut(x,0)=0,

boundary conditions:

u(0,t)=0,u(1,t)=1+cosh(t),

and exact solution:

u(x,t)=x+x2cosh⁡(t).

Similarly, the proposed neural architecture has been employed for approximating solution of this initial-boundary problem on square Ω = [0, 1] × [0, 0.5] for m = n = 5 and m′ = n′ = 10. The obtained numerical results are presented in Table 2. We are now allowed to claim that the combination method proposed in this paper can be applied to accurately approximate the unknown functions to any desired degree of preciseness. In particular, thanks to this unique characteristic of artificial neural networks, the algorithmic power series method can be converted into an iterative non-algorithmic one. Here, if we consider an arbitrarily large number of iterations, the proposed boosting method will be able to approximate the unknown function with high precision.

4 Conclusion

Certain properties of artificial neural networks are typically configured to distinguish the networks based on iterative approach from other classical numerical methods. In this research, a combination of ANNs approach and power series methods, was used for numerical solution of two special types of partial differential equations. Wave equations and heat equations are considered as two main boundary-initial value partial differential equations which have played pivotal part in modeling physics phenomena. The proposed approach could convert solving a differential problem into related optimization minimizing problem. This work combined initial and boundary conditions of the problem, which could be easily modeled with suitable network architecture. Discretizing the differential domain and then using the back-propagation learning algorithm lead to solveing the optimization problem for the unknown series of coefficients. The validity of our method was based on the supposition that the convergence rate quickly rises by increasing the number of node points and learning steps. However, the initial values for the network parameters had a considerable impact. The learning rate and momentum constant were sensitive tools that were considered as convergence speed control parameters. Inappropriate choices for these parameters led to a lack of convergence or an excessive increase in the number of repeating steps. To make a better description of the offered technique, one numerical example was presented with computer simulations. Also, comparison of numerical results with exact solutions and those of another classical method has helped us to precisely understand this exercise. The achieved results support our claim that the designed neural architecture gives better convergent approximation without any restrictive assumptions. However, most of the equations belong to mathematical applications in real-world problems; therefore they require complex solutions. According to the numerical results obtained from two examples, it was natural to claim that the proposed procedures were valid and possessed unique properties along with high efficiency. The proposed method was more efficient than other methods. With a little care in the performance of this method, it can be easily concluded that our combination technique can be classified in row of non-algorithmic methods. Despite having control levers such as learning rate, momentum constant or variety of learning algorithms and cost functions increase the accuracy in determining the mathematical mystery. It is obvious that most classical methods are not available to solve a variety of complex mathematical problems. In many cases, these non-algorithmic methods can solve difficult mathematical problems. Research in this area can provide great benefits to the related fields, while extension of the fractional partial differential equations can be a milestone for the near future research.

Table 3

Absolute error comparison for r = 10000

t = 0t = 0.1
– – – – – – – – – – – – – –– – – – – – – – – – – – – –– – – – – – – – – – – – – –– – – – – – – – – – – – – –
xTE methodANNs approachTE methodANNs approach
0.1001.3891e – 119.8640e – 12
0.2005.5565e – 115.8647e – 11
0.3001.2502e – 103.8402e – 11
0.4002.2226e – 102.8223e – 11
0.5003.4728e – 104.7073e – 11
0.6005.0009e – 106.9314e – 11
0.7006.8068e – 103.8425e – 11
0.8008.8905e – 105.8014e – 11
0.9001.1252e – 097.3592e – 11
t = 0.2t = 0.3
– – – – – – – – – – – – – –– – – – – – – – – – – – – –– – – – – – – – – – – – – –– – – – – – – – – – – – – –
xTE methodANNs approachTE methodANNs approach
0.18.8952e – 102.7518e – 111.0141e – 082.7468e – 11
0.23.5581e – 092.8677e – 114.0565e – 084.3099e – 11
0.38.0057e – 095.0046e – 119.1272e – 084.4029e – 11
0.41.4232e – 083.9617e – 111.6226e – 073.6418e – 11
0.52.2238e – 084.8135e – 112.5353e – 073.9214e – 11
0.63.2023e – 086.0175e – 113.6509e – 074.7109e – 11
0.74.3587e – 086.1482e – 114.9692e – 076.9243e – 11
0.85.6930e – 084.0841e – 116.4904e – 077.2390e – 11
0.97.2051e – 083.6940e – 118.2144e – 077.8814e – 11
t = 0.4t = 0.5
– – – – – – – – – – – – – –– – – – – – – – – – – – – –– – – – – – – – – – – – – –– – – – – – – – – – – – – –
xTE methodANNs approachTE methodANNs approach
0.15.7052e – 083.8425e – 112.1799e – 073.9627e – 11
0.22.2821e – 073.1208e – 118.7194e – 073.0281e – 11
0.35.1347e – 076.0172e – 111.9619e – 063.0127e – 11
0.49.1283e – 074.8466e – 113.4878e – 064.8692e – 11
0.51.4263e – 061.4370e – 115.4496e – 066.3484e – 11
0.62.0539e – 061.3692e – 117.8475e – 066.9107e – 11
0.72.7955e – 068.2410e – 111.0681e – 058.1123e – 11
0.83.6513e – 067.3596e – 111.3951e – 059.3271e – 11
0.94.6212e – 067.1104e – 111.7657e – 051.3504e – 10

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Received: 2016-8-31
Accepted: 2017-6-9
Published Online: 2017-12-29

© 2017 A. Jafarian and D. Baleanu

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/phys-2017-0135
Keywords for this article
wave-like and heat-like equations; bivariate power series polynomial; artificial neural network; criterion function; back-propagation learning algorithm
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BY-NC-ND 4.0

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  59. Regular Articles
  60. Measurement problem and local hidden variables with entangled photons
  61. Regular Articles
  62. Compression of hyper-spectral images using an accelerated nonnegative tensor decomposition
  63. Regular Articles
  64. Fabrication and application of coaxial polyvinyl alcohol/chitosan nanofiber membranes
  65. Regular Articles
  66. Calculating degree-based topological indices of dominating David derived networks
  67. Regular Articles
  68. The structure and conductivity of polyelectrolyte based on MEH-PPV and potassium iodide (KI) for dye-sensitized solar cells
  69. Regular Articles
  70. Chiral symmetry restoration and the critical end point in QCD
  71. Regular Articles
  72. Numerical solution for fractional Bratu’s initial value problem
  73. Regular Articles
  74. Structure and optical properties of TiO2 thin films deposited by ALD method
  75. Regular Articles
  76. Quadruple multi-wavelength conversion for access network scalability based on cross-phase modulation in an SOA-MZI
  77. Regular Articles
  78. Application of ANNs approach for wave-like and heat-like equations
  79. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  80. Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory
  81. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  82. A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
  83. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  84. Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
  85. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  86. On the libration collinear points in the restricted three – body problem
  87. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  88. Research on Critical Nodes Algorithm in Social Complex Networks
  89. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  90. A simulation based research on chance constrained programming in robust facility location problem
  91. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  92. A mathematical/physics carbon emission reduction strategy for building supply chain network based on carbon tax policy
  93. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  94. Mathematical analysis of the impact mechanism of information platform on agro-product supply chain and agro-product competitiveness
  95. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  96. A real negative selection algorithm with evolutionary preference for anomaly detection
  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  164. Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  166. The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  182. Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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