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BY 4.0 license Open Access Published by De Gruyter September 25, 2019

Approximate method for solving strongly fractional nonlinear problems using fuzzy transform

  • Mohamad Adabitabar Firozja and Bahram Agheli EMAIL logo
From the journal Nonlinear Engineering

Abstract

In this research work, we have shown that it is possible to use fuzzy transform method (FTM) for approximate solution of strongly fractional nonlinear problems. In numerical methods, in order to approximate a function on a particular interval, only a restricted number of points are employed. However, what makes the F-transform preferable to other methods is that it makes use of all points in this interval. The comparison of the time used in minutes is given for two derivatives Caputo derivative and Caputo-Fabrizio derivative.

1 Introduction

Fractional arithmetic and fractional differential equations appeared in many sciences, including medicine [1], economics [2], dynamical problems [3, 4], chemistry [5], chaotic systems [6], mathematical physics [7, 8, 9, 10, 11], traffic model [12], entropy [13] and fluid flow [14] and so on. Scholars and researchers are invited to check books that have been written to take advantage of fractional arithmetic [15, 16].

Many researchers have used numerical methods for the purpose of solving the fractional Riccati differential equations (FRDEs) and the fractional Bratu differential equations (FBDEs).

In this research work, we have for the first time shown that it is possible to use F-transform method (FTM) to tackle with FRDEs and FBDEs of the following forms.

  1. Fractional Riccati differential equations (FRDEs)

    Dαu(t)i=02pi(t)ui(t)=0,0<α1,r0<tT,u(0)=u0,(1.1)

    where t ∈ ℝ, pi(t), i = 0, 1, 2 are constant functions.

  2. Fractional Bratu differential equations (FBDEs)

    Dαu(t)λexp(u(t))=0,1<α2,0<tT,u(0)=u0,ut(0)=u0,(1.2)

    where λ > 0 and t ∈ ℝ, pi(t), i = 0, 1, 2 are constant functions.

The operator Dα denotes the Caputo’s derivative [16] of order α

Dαu(t)=1Γ(nα)0t(ts)α1u(n)(s)ds,t>a,n1<αn,nN,(1.3)

or Caputo-Fabrizio’s derivative [17] of order α

Dtn+αu(t)=T(α)1αatexp((st)α1α)u(n+1)(s)ds,t>a,n<αn+1,nZ+,(1.4)

in which, T(α) is called, the normalization function featuring T (0) = T (1) = 1.

Historically, a special case of this differential equation by James Bernoulli (1654-1705) and then by Count Jacopo Francesco Riccati (1676-1754) was introduced and evaluated. On the importance and motivation for this differential equation, it should be noted that it has a key role in many of the physical phenomena and other sciences. Such applications can include control systems, robust stabilization, network synthesis, diffusion problems, optimal filtering, controls, stochastic theory, financial mathematics, optimal control, river flows, robust stabilization, network synthesis and financial mathematics dynamic games, linear systems with Markovian jumps, stochastic control, econometric models and invariant embedding noted that the use of the Riccati differential equation [18, 19, 20, 21, 22, 23, 24, 25]. Of other uses, the one dimensional static Schrödinger equation [26] and the travelling wave solutions of a nonlinear partial differential equation [27] are noteworthy with the Riccati differential equation featuring fractional derivatives.

On the importance and motivation for Bratu differential equation, it should be noted that it has a key role in many of the physical phenomena, chemical models and other sciences. Such applications can include model of thermal reaction process, the fuel ignition model of the thermal combustion theory, the Chandrasekhar model of the expansion of the universe, radiative heat transfer nanotechnology and chemical reaction theory [28, 29, 32, 33]

The FTM has recently been utilized by authors in [34, 35, 36] to find approximate solution of the first order fuzzy differential equations and two-point boundary value problems. Along the same line of research, Chen and his associates in [37] have established an algorithm to gain the numerical solutions of second order primary amount problems.

It must be pointed out here that researchers have utilized disparate schemes to solve FRDEs during the last two decades. We can refer to familiar methods, including differential transform method [38], Adomian’s decomposition method [39], homotopic perturbation method [40], variational iteration method [41], homotopic analysis method [41] and etc [42, 43, 44]. For numerical solution of FBDEs we can point to homotopic perturbation method [46], optimal homotopy asymptotic method [47] and variational iteration technique [48] and etc. Scholars and researchers are invited to study other numerical solutions in [49, 50, 51, 52, 53, 54]

2 Discretization of the fractional derivative

Assume that u(t) is the solution to equations (1.1) and (1.2). To calculate the approximation of u(t), we use the discretization of the Caputo derivative and Caputo-Fabrizio derivative with the assumption τ = tj+1tj and tj = a + jτ, j = 0, 1, 2, ⋯.

2.1 Discretization of the Caputo derivative

Utilizing the approximation for the Caputo derivative [56] of Eq. (1.3) we have:

Dαu(tk+1)1ταΓ(2α)j=0k(u(tj+1)u(tj))(kj+1)1α(kj)1α,(2.1)

in which 0 < α ≤ 1, u(t0) is known and

Dαu(tk+1)1ταΓ(3α)j=0k(u(tj+1)2u(tj)+u(tj1))×(kj+1)2α(kj)2α,(2.2)

in which 1 < α ≤ 2, u(t0) and u(t0) are known and and u(t–1) = u(t0) − τu(t0).

2.2 Discretization of the Caputo-Fabrizio derivative

Utilizing the approximation for the Caputo-Fabrizio derivative [57] of Eq. (1.4) we have:

Dαu(tk+1)1ατj=0k(u(j+1)u(j))×exp((ατ)(kj)1α)exp((ατ)(kj+1)1α),(2.3)

in which 0 < α ≤ 1, u(t0) is known and

Dαu(tk+1)1ατ2j=0k(u(j1)+u(j+1)2u(j))×exp(ατ(kj)1α)exp(ατ(kj+1)1α),(2.4)

in which 1 < α ≤ 2, u(t0) and u(t0) are known and u(t–1) = u(t0) − τu(t0).

3 Fuzzy partition and fuzzy transform

In this section, only the main definitions of F-transform to be utilized in the subsequent sections of numerical implementations will be outlined.

Definition 3.1

[55] Presuming that for n ≥ 2, a = t1 < t2 < ⋯ < tn–1 < tn = b be specified nodes, we express that fuzzy sets B1, ⋯, Bn defined on [a, b] with their membership functions B1(t), ⋯, Bn(t), form a fuzzy partition of [a, b] if they meet the following properties:

  1. Bk of [a, b] to [0, 1] is continuous, k=1nBk(t) = 1 for all t ∈ [a, b] and Bk(tk) = 1, k = 1, 2, ⋯, n.

  2. Bk(t) = 0 if t ∉ (tk–1, tk+1), with t0 = a and tn+1 = b,

  3. On subinterval [tk–1, tk+1], for k = 2, ⋯, n − 1, Bk(t), certainly is an increasing function on [tk–1, tk] and decreasing function on [tk, tk+1].

    The membership functions B1, B2, ⋯, Bn are named basic functions (BFs).

    The next formulas give the standard display of such triangular membership functions:

    B1(t)=1tt1h1,t1tt20,otherwise,Bk(t)=ttk1hk1,tk1ttk1ttkhk,tkttk+1,k=2,3,,n1,0,otherwise,Bn(t)=ttn1hn1,tn1ttn,0,otherwise.(3.1)

    The formulas that follow for k = 2, ⋯, n − 1 give the standard display of such sinusoidal membership functions:

    B1(t)=0.51+cosπh(tt1),t1tt20,otherwise,Bk(t)=0.51+cosπh(ttk),tk1ttk+1,k=2,3,,n1,0,otherwise,Bn(t)=0.51+cosπh(ttn),tn1ttn0,otherwise,(3.2)

    in which hk = tk+1tk for k = 1, ⋯, n − 1. It can be stated that fuzzy partition of [a, b], is uniform if tk+1tk = h = ban1 and two additional properties coincide:

  4. Bk(tkt) = Bk(tk + t), for all t ∈ [0, h], for k = 2, ⋯, n − 1,

  5. Bk(t) = Bk–1(th) and Bk+1(t) = Bk(th), for k = 2, ⋯, n − 1, and t ∈ [tk, tk+1].

Definition 3.2

[55] Let f be any function belonging to C([a, b]) and B1, B2, ⋯, Bn, be the BFs which buildup a fuzzy partition of [a, b]. We define the n-tuple [F1, F2, ⋯, Fn] of real numbers given by

Fk=abf(t)Bk(t)dtabBk(t)dt,k=1,2,,n,(3.3)

as the F-transform of f in relation to B1, B2, ⋯, Bn.

Definition 3.3

[55] Let [F1, F2, ⋯, Fn] be the F-transform of function f relative to BFs, B1, B2, ⋯, Bn. Then,

fn(t)=k=1nFkBk(t),

is named the inverse F-transform (IFT) of function f on [a, b].

Theorem 3.4

[55] Let f be a continuous function on [a, b] and B1, B2, ⋯, Bn be the BFs which form a fuzzy partition of [a, b]. Then, the kth component of the integral F-transform signified over [f(a), f(b)], gives the minimum to the function

ϕ(y)=abf(t)y2Bk(t)dt.

Lemma 3.5

[55] (Convergence) Let f be a continuous function on [a, b]. Thus, for any ϵ > 0, there exist nϵ and a fuzzy partition B1, ⋯, Bnϵ of [a, b] such that for all t ∈ [a, b]

f(t)fnϵ(t)ϵ.(3.4)

4 Description of the new approach

Let u(t) be the continuous solution of (1.1) on [0, T]. Also, U1, ⋯, Un of F-transform u(t), calculated by using BFs B0, B1, ⋯, Bn in [0, T] regarding (3.1) with tj+1tj = τ which are uniform fuzzy partitions. Now with applying IFT on the function u(t) give the approximation un (x) as according to:

un(t)=k=0nUkBk(t),t0,T.(4.1)

Hence for approximate solution, we can calculated Uk for k = 0, 1, 2, ⋯, n, where Uk, are not F-transform of u and must be calculated.

In the next proposition the discretization of the Caputo derivative for un (t) for Eqs.(2.1), (2.2), (2.3) and (2.4) are presented.

Proposition 4.1

With substituting un(t) in Eqs.(2.1), (2.2), (2.3) and (2.4), we will have the next equations, respectively:

Dαun(tk+1)1ταΓ(2α)j=0k(Uj+1Uj)(kj+1)1α(kj)1α,0<α1,(4.2)
Dαun(tk+1)1ταΓ(3α)j=0k(Uj+12Uj+Uj1)×(kj+1)2α(kj)2α,1<α2,(4.3)
Dαun(tk+1)1ατj=0k(Uj+1Uj)×exp((ατ)(kj)1α)exp((ατ)(kj+1)1α),0<α1,(4.4)
Dαun(tk+1)1ατ2j=0k(Uj+12Uj+Uj1×exp((ατ)(kj)1α)exp((ατ)(kj+1)1α),1<α2,(4.5)

where u(t0) and u(t0) are known of initial conditions, U0 = u(t0) and U–1 = u(t0) − τu(t0).

4.1 Approximate solution of FRDEs

In order to gain the approximate solution of the problem (1.1), we use of un(t), hence

Dαun(t)=i=02pi(t)uni(t),0<α1,0<tT,(4.6)

and by putting t = tk+1, we have

Dαun(tk+1)=i=02pi(tk+1)uni(tk+1),0<α1,k=0,1,,n1.(4.7)

  1. Considering Caputo’s derivative:

    using Eq.(4.2), Eq.(4.7) convert to the following form

    1ταΓ(2α)j=0k(Uj+1Uj)(kj+1)1α(kj)1α
    =i=02pi(tk+1)Uk+1i,k=0,1,2,,n1.(4.8)
  2. Considering Caputo-Fabrizio derivative:

    using Eq.(4.4), Eq.(4.7) convert to the following form

    1ατj=0k(Uj+1Uj)exp((ατ)(kj)1α)exp((ατ)(kj+1)1α)=i=02pi(tk+1)Uk+1i,(4.9)

    for k = 0, 1, 2, ⋯, n − 1.

Now, using the boundary condition, we can calculate U1, U2, ⋯, Un by recursive equation and then by IFT gain the approximate solution u(t) ≈ un(t) for Eq.(1.1).

An algorithm for approximation of FRDEs by this method stated in the next Algorithm.

Algorithm 1

An approximation algorithm for FRDEs

  1. Input p0(t), p1(t), p2(t), U0 = u(0), n and T.

  2. Set τTn.

  3. Locate tkkτ, k = 0, 1, 2, ⋯, n.

  4. Choose sinusoidal BFsBk(t) for k = 0, 1, 2, ⋯, n.

    1. With Caputo derivative, set recursive equation

      1ταΓ(2α)j=0k(Uj+1Uj)(kj+1)1α(kj)1α=i=02pi(tk+1)Uk+1i,
    2. With Caputo-Fabrizio derivative, set recursive equation

      1ατj=0k(Uj+1Uj)exp((ατ)(kj)1α)exp((ατ)(kj+1)1α)=i=02pi(tk+1)Uk+1i.

      for k = 0, 1, 2, ⋯, n − 1.

  5. Calculate every Uk, k = 1, 2, ⋯, n of an equation of degree two. ([U0, U1, U2, ⋯, Un] are F-transform.)

  6. The approximate solution with IFT is

    un(t)=k=0nUkBk(t).

4.2 Approximate solution of FBDEs

In order to gain the approximate solution of the problem (1.2), we use of un(t), hence

Dαun(t)=λexp(un(t)),1<α2,0<tT,(4.10)

and by putting t = tk+1, we have

Dαun(tk+1)=λexp(un(tk+1)),λ>0,1<α2,k=0,1,,n1.(4.11)

  1. Considering Caputo’s derivative:

    using Eq.(4.3), Eq.(4.11) convert to the following form

    1ταΓ(3α)j=0k(Uj+12Uj+Uj1)(kj+1)2α(kj)2α=λexp(Uk+1),k=0,1,2,,n1.(4.12)
  2. Considering Caputo-Fabrizio derivative:

    using Eq.(4.5), Eq.(4.11) convert to the following form

    1ατ2j=0k(Uj+12Uj+Uj1)exp((ατ)(kj)1α)exp((ατ)(kj+1)1α)=λexp(Uk+1),(4.13)

    in which k = 0, 1, 2, ⋯, n − 1.

Now, using the boundary condition, we can calculate U1, U2, ⋯, Un by recursive equation and then by IFT gain the approximate solution u(t) ≈ un(t) for Eq.(1.2).

An algorithm for approximation of FBDEs by this method stated in the next Algorithm.

Algorithm 2

An approximation algorithm for FBDEs

  1. Input U0 = u(0) and ut(0) = u0, n and T.

  2. Set τTn and U–1U0τu0.

  3. Locate tkkτ, k = 0, 1, 2, ⋯, n.

  4. Choose sinusoidal BFsBk(t) for k = 0, 1, 2, ⋯, n.

    1. With Caputo derivative, set recursive equation

      1ταΓ(3α)j=0k(Uj+12Uj+Uj1)×(kj+1)2α(kj)2α=λexp(Uk+1).
    2. With Caputo-Fabrizio derivative, set recursive equation

      1ατ2j=0k(Uj+12Uj+Uj1)exp((ατ)(kj)1α)exp((ατ)(kj+1)1α)=λexp(Uk+1),

      for k = 0, 1, 2, ⋯, n − 1.

  5. Calculate every Uk, k = 1, 2, ⋯, n of an equation of degree one. ([U0, U1, U2, ⋯, Un] are F-transform.)

  6. The approximate solution with IFT is

    un(t)=k=0nUkBk(t).

5 Examples

Now in this section, we present various examples for illustrate FTM for FRDEs and FBDEs. In all these examples, we used of mathematical software Mathematica.

Example 5.1

For the first example, we propose the FRDEs [43]:

Dtαu(t)=1u2(t),0<t<1,0<α1,(5.1)

with the precise solution u(t)=exp(2t)1exp(2t)+1 for α = 1 and the primary condition:

u0=u(0)=0.(5.2)

Following the FTM, according to what was formulated and presented in section 4 for Eqs.(5.1)-(5.2), we can calculate U1, U2, …, Un and then gain the approximate solution un(t) of (5.1).

Table 1 shows comparison betwixt the exact and the approximation solution (5.1) with F-transform of test example 5.1 for different values of α and t, n = 500, τ = 0.002, featuring Caputo and Caputo-Fabrizio derivative.

Table 1

The exact and approximate result of test example 5.1 featuring various values of α, with Caputo and Caputo-Fabrizio derivative.

CaputoCaputo-Fabrizio
tα = 0.5α = 0.75α = 1.0Exactα = 0.5α = 0.75α = 1.0Exact
0.00.00.00.00.00.00.00.00.0
0.20.3346260.2609410.1974950.1973750.3346150.2609320.1974370.197375
0.40.4984660.4426380.3799720.3799490.4984590.4426290.3798950.379949
0.60.6045880.5777810.5369210.537050.6045780.5777800.5368470.53705
0.80.6774290.676930.663830.6640370.6774180.676870.663770.664037
1.00.7295030.7491040.7614070.7615940.7295020.7491030.761370.761594

Comparison of exact and approximate solution can be seen for equations with different values of α, n = 500, τ = 0.002 and various values of t, in Figure 1 with Caputo derivative and in Figure 2 with Caputo-Fabrizo derivative.

Fig. 1 Comparison betwixt the exact and the approximation solution with F-transform of test example 5.1 for n = 500, τ = 0.002 and various values of t and α with Caputo-Fabrizio derivative.
Fig. 1

Comparison betwixt the exact and the approximation solution with F-transform of test example 5.1 for n = 500, τ = 0.002 and various values of t and α with Caputo-Fabrizio derivative.

Fig. 2 Comparison betwixt the exact and the approximation solution with F-transform of test example 5.1 for value of n = 500, τ = 0.002 and different values of α and t with Caputo derivative.
Fig. 2

Comparison betwixt the exact and the approximation solution with F-transform of test example 5.1 for value of n = 500, τ = 0.002 and different values of α and t with Caputo derivative.

Table 2 represents the present method for α = 1 and the achieved results of homotopy perturbation method ( HPM), Adomian decomposition method (ADM) [30] and optimal homotopy asymptotic method (OHAM) [31].

Table 2

Comparison of the numerical solutions of the equation in example 5.1 with α = 1.

FTM
tHPMADMOHAMCaputoCaputo-FabrizioExact
0.20.1973750.1973750.1974020.1974370.1974370.197375
0.40.3799430.3799480.3800650.3799720.3798950.379949
0.60.5368570.5370490.5371480.5369210.5368470.53705
0.80.6617060.6640370.6640490.663830.663770.664037
1.00.74603180.7616220.7616340.7614070.761370.761594

Example 5.2

For the second example, we offer the FBDEs [32]:

Dtαu(t)2exp(u(t))=0,1<α2(5.3)

including the primary condition

u0=u(0)=0,u0=ut(0)=0.(5.4)

The unknown coefficient U1, U2, ⋯, Un with due attention to the FTM, according to section 4 for Eqs.(5.3)-(5.4) are calculated.

Comparison of exact and approximate solution can be seen in Table 3 for equations with n = 500, τ = 0.002 and various values of t and α, featuring Caputo fractional derivative.

Table 3

The exact and approximate result of test example 5.2 featuring various values of α, with Caputo and Caputo-Fabrizio derivative.

CaputoCaputo-Fabrizio
tα = 0.5α = 0.75α = 1.0Exactα = 0.5α = 0.75α = 1.0Exact
0.00.00.00.00.00.00.00.00.0
0.20.08596060.0592720.04043690.04026950.08596050.0592710.04043680.0402695
0.40.2925980.2198670.1648020.1644580.2925970.2198660.1648020.164458
0.60.6300750.4906160.3844550.383930.6300760.4906150.3844540.38393
0.81.159950.9070690.7236580.7227811.159950.9070700.7236560.722781
1.01.544871.193731.233151.231251.544871.193731.233151.23125

Figure 3 and Figure 4 shows comparison betwixt the exact and the approximation solution (5.1) with F-transform of test example 5.1 for various values of α, n = 500, τ = 0.002, respectively, with Caputo and Caputo-Fabrizio fractional derivative.

Fig. 3 Comparison betwixt the exact and the approximation solution with F-transform of test example 5.2 for different values of α, n = 500, τ = 0.002 and various values of t, with Caputo fractional derivative.
Fig. 3

Comparison betwixt the exact and the approximation solution with F-transform of test example 5.2 for different values of α, n = 500, τ = 0.002 and various values of t, with Caputo fractional derivative.

Fig. 4 Comparison betwixt the exact and the approximation solution with F-transform of test example 5.2 for value of n = 500, τ = 0.002 and various values of α and t, with Caputo-Fabrizio derivative.
Fig. 4

Comparison betwixt the exact and the approximation solution with F-transform of test example 5.2 for value of n = 500, τ = 0.002 and various values of α and t, with Caputo-Fabrizio derivative.

Toward α = 2, the solution that we have gained is in accordance with the precise solution u(t) = − 2 log (cos (t)).

Example 5.3

For the second example, we offer the FBDEs [32]:

Dtαu(t)+2exp(u(t))=0,1<α2(5.5)

including the primary condition

u0=u(0)=0,u0=ut(0)=0.(5.6)

Toward α = 2, the solution that we have gained is in accordance with the precise solution u(t) = − 2. log (0.848338 cosh (1.17878 (t − 0.5))).

Table 4 represents the present method for α = 2 and the achieved results of Laplace transform method (LTM), decomposition method (DM) and B-spline method (BSM) [58].

Table 4

Comparison of the numerical solutions of the equation in example 5.3 with α = 2.

FTM
tLTMDMBSMCaputoCaputo-FabrizioExact
0.50.3193530.3359370.3288960.327580.3275910.328952
0.60.3041600.3183360.3150360.3134990.3134830.315089
0.70.2619460.2679910.2738340.2721730.2720870.273879
0.80.1940410.1917440.2063860.2046660.2045170.206419
0.90.1035370.0991930.1143930.112620.1124840.114411

In this method, by increasing the amount n and decreasing the amount τ, a more accurate answer can be achieved. The time that the CPU is used in minutes for FRDEs with α = 1 and FBDEs with α = 2 featuring Caputo derivative and Caputo-Fabrizio derivative in difference, τ = 0.002, n = 50 and n = 500 is shown in Table 5. Baleanu et.al in [59] in a non-difference state compared the Caputo derivative and Caputo-Fabrizio derivative in terms of run-time in seconds.

Table 5

Duration used in minutes.

FRDEsFBDEs
CaputoCaputo-FabrizioCaputoCaputo-Fabrizio
n = 500.03697920.04036460.09140630.09375
n = 5005.616935.621617.070317.65938

6 Conclusion

We have successfully applied FTM to obtain approximate solution of the FRDEs and FBDEs. The result indicate that a few iteration of FTM will result in some useful solutions. Finally, it should be added that the suggested technique has the potentials to be practical in solving other similar nonlinear and linear problems in partial differential equations featuring fractional derivative.

Acknowledgement

We are very much indebted to Prof. Irina Perfilieva for constructive comments and helpful suggestions which led to improved presentation and quality of this paper.

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Received: 2018-08-19
Revised: 2018-11-16
Accepted: 2019-01-28
Published Online: 2019-09-25

© 2020 M. Adabitabar Firozja and B. Agheli, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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