Analytical and approximate solutions of Fractional Partial Differential-Algebraic Equations

Hatıra Günerhan 1  and Ercan Çelik
  • 1 Mathematics Department, Faculty of Education, Kars, Turkey
Hatıra Günerhan and Ercan Çelik

Abstract

In this paper, we have extended the Fractional Differential Transform method for the numerical solution of the system of fractional partial differential-algebraic equations. The system of partial differential-algebraic equations of fractional order is solved by the Fractional Differential Transform method. The results exhibit that the proposed method is very effective.

1 Introduction

In the past several years ago, various methods have been proposed to obtain the numerical solution of partial differential-algebraic equations [2], [7], [11], [12], [13], [14], [15], [16]. In this study, we consider the following system of partial differential-algebraic equations of fractional order

ADαtv(t,x)+BLxv(t,x)+Cv(t,x)=f(t,x),
Where α is a parameter describing the fractional derivative and t ∈ (0, te), 0 < α ≤ 1 and x ∈ (−l, l) ⊂ R, A, B, CRn×n, are constant matrices, u, f : [0, te] × [−l, l] → Rn. The purpose of this paper is to consider the numerical solution of FPDAEs by using Fractional Differential Transform Method.

2 Basic Definitions

We give some basic definitions and properties of the fractional calculus theory which are used further in this paper.

Definition 1

A real function f (x), x > 0 is said to be in the space Cµ, µεR if there exists a real number P > µ such that f (x) = xp f1(x), where f1(x)εC[(0, ∞). Clearly Cµ < Cβ if µ < β.

Definition 2

A function f (x), x > 0 is said to be in the space Cμm , mεN ∪ {0} if f(m)Cµ.

Definition 3

The Riemann-Liouville fractional integral operator of the order α > 0 of a function, fCµ, µ ≥ −1 is defined as:

(Jaαf)(x)=1Γ(α)ax(xτ)α1f(τ)dτ,x>a,
(Ja0f)(x)=f(x).

Properties of the operator Jα can be found in (Caputo, 1967), we mention only the following:

For fCµ, µ ≥ −1, α, β ≥ 0, and γ > −1.

(JaαJaβf)(x)=(Jaα+βf)(x),
(JaαJaβf)(x)=(JaβJaαf)(x)
Jaαxγ=Γ(γ+1)Γ(α+γ+1)xα+γ.

Definition 4

The fractional derivative of f (x) in the Caputo sense is defined as

(Dαaf)(x)=(JmαaDmf)(x)=1Γ(ma)ax(xt)mα1f(m)(t)dt,
for m − 1 < α < m, mN, x > 0,.

Lemma 1

If −1 < α < m, mN and µ ≥ −1, then

(JaαDaαf)(x)=f(x)k=0m1fk(a)((xa)kk!),a0
(DaαJaαf)(x)=f(x)

3 fractional Two-Dimensional Differential Transform Method

Differential Transform Method (DTM) is an analytic method based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional high order Taylor series method requires symbolic computation. However, the FDTM obtains a polynomial series solution using an iterative procedure. The proposed method is based on the combination of the classical two-dimensional FDTM and generalized Taylor’s Table 1 formula. Consider a function of two variables u(x, y) and suppose that it can be represented as a product of two single-variable functions, that is, u(x, y) = f(x)g(y) based on the properties of fractional two-dimensional differential transform [1], [3], [4], [5], [6], [8], [9], [10], the function u(x, y) can be represented as:

Table 1

The operations for the two-dimensional differential transform method

Transformed functionOriginal function
Uα,β = Vα,β +Wα,βu(x,y) = v(x,y)w(x,y)
Uα,β = λVα,βu(x,y) = λv(x,y)
Uα,β(k,h)=r=0ks=0hVα,β(r,hs)Wα,β(kr,s)u(x,y) = v(x,y)w(x,y)
Uα,β(k,h)=δ(kn)δ(hm)={1,k=n,h=m0,kn,hmu(x,y) = (xx0) (yy0)
Uα,β(k,h)=Γ(α(k+1)+1)Γ(αk+1)Vα,β(k+1,h)Vα,β(k+ 1, h)u(x,y)=Dx0αv(x,y)
Uα,β(k,h)=Γ(β(h+1)+1)Γ(βh+1)Vα,β(k,h+1)Vα,β(k,h+ 1)u(x,y)=Dy0βv(x,y)
Uα,β(k,h)=Γ(α(k+1)+1)Γ(αk+1)Γ(β(h+1)+1)Γ(βh+1) . Vα,β(k+ 1, h+ 1), 0< α,β ≤ 1u(x,y)=Dx0αDy0βv(x,y)
u(x,y)=k=0Fα(k)(xx0)kαh=0Gβ(h)(yy0)hβ=k=0h=0Uα,β(k,h)(xx0)kα(yy0)hβ,
Where 0 < α, β ≤ 1, Uα,β (k, h) = Fα(k)Gβ (h), is called the spectrum of u(x, y). The fractional two-dimensional differential transform of the function u(x, y) is given by
Uα,β(k,h)=1Γ(αk+1)Γ(βh+1)[(Dx0α)k(Dy0β)hu(x,y)](x0,y0).
Where (Dx0α)k=Dx0αDx0αDx0αk

In the case of α = 1 and β = 1 the Fractional two-dimensional differential transform (9) reduces to the classical two-dimensional differential transform. Let Uα,β (k, h), wα,β (k, h) and Vα,β (k, h) are the differential transformations of the functions u(x, y), w(x, y) and v(x, y), from Equations(9) and (10) , some basic properties of the two-dimensional differential transform are introduced in Table 1 .

Then, the fractional differential transform (10) becomes;

Uα,β(k,h)=1Γ(αk+1)Γ(βh+1)[Dx0αk(Dy0β)hu(x,y)](x0,y0),

4 Numerical example

Here, the fractional differential transform method will be applied for solving the fractional partial differential-algebraic equation.

Example 2

Consider the fractional partial differential-algebraic equation

(011211000)Dtαv+(000000001)vxx+(000010001)v=f,
x ∈ [−0.5, 0.5], t ∈ [0,1].

With initial condition

v1(x,0)=x2,v1t(x,0)=x2,v2(x,0)=x2,v2t(x,0)=x22,v3(x,0)=0,v3t(x,0)=x2,
where
f1(x,t)=12x2e12t+x2cos(t),f2(x,t)=2x2et12x2e12tx2cos(t),f3(x,t)=2sin(t)+x2sin(t),
with the exact solution
v(x,t)=(x2etx2et2x2sin(t)),

Equivalently, equation (12) can be written as

(011211000)(Dtαv1Dtαv2Dtαv3)+(000000001)(v1xxv2xxv3xx)+(000010001)(v1v2v3)=(f1f2f3),
Dtαv2+Dtαv3=f1,2Dtαv1Dtαv2Dtαv3v2=f2,v3xx+v3=f3.

By using the DTM in equation (17) , we obtain

V2(k,h+1)Γ(α(h+1)+1)Γ(αh+1)+V3(k,h+1)Γ(α(h+1)+1)Γ(αh+1)=F1(k,h),2V1(k,h+1)Γ(α(h+1)+1)Γ(αh+1)V2(k,h+1)Γ(α(h+1)+1)Γ(αh+1)+V3(k,h+1)Γ(α(h+1)+1)Γ(αh+1)V2(k,h)=F2(k,h),(k+2)(k+1)V3(k+2,h)+V3(k,h)=F3(k,h),
from the initial condition (13) , we have
V1(k,1)=δ(k2)={1k=2,0k2,V2(k,1)=12δ(k2)={12k=2,0k2,V3(k,0)=0,k=0,1,,V1(k,0)=V2(k,0)=V3(k,1)=δ(k2)={1k=2,0k2,.

By using the differential inverse reduced transform of V1(k, h), V2(k, h) and V3(k, h), we get

v1(x,t)=[11Γ(α+1)tα+6Γ(α+1)+28Γ(2α+1)t2α+54Γ(2α+1)+12Γ(α+1)96Γ(3α+1)t3α+50Γ(3α+1)54Γ(2α+1)192Γ(4α+1)t4α+330Γ(4α+1)+40Γ(3α+1)7680Γ(5α+1)t5α+66Γ(5α+1)+310Γ(4α+1)15360Γ(6α+1)t6α+1806Γ(6α+1)+84Γ(5α+1)1290240Γ(7α+1)t7α+770Γ(7α+1)1806Γ(6α+1)2580480Γ(8α+1)t8α+9234Γ(8α+1)+144Γ(7α+1)371589120Γ(9α+1)t9α+1026Γ(9α+1)+9198Γ(8α+1)743178240Γ(10α+1)t10α+45078Γ(10α+1)+220Γ(9α+1)163499212800Γ(11α+1)t11α(+]x2,v2(x,t)=[1+12Γ(α+1)2Γ(α+1)tα+Γ(α+1)4Γ(2α+1)t2α+27Γ(2α+1)+8Γ(3α+1)48Γ(3α+1)t3α+Γ(3α+1)96Γ(4α+1)t4α+155Γ(4α+1)32Γ(5α+1)3840Γ(5α+1)t5α+Γ(5α+1)7680Γ(6α+1)t6α+903Γ(6α+1)+128Γ(7α+1)645120Γ(7α+1)t7α+Γ(7α+1)1290240Γ(8α+1)t8α+4599Γ(8α+1)512Γ(9α+1)185794560Γ(9α+1)t9α+Γ(9α+1)371589120Γ(10α+1)t10α+22539Γ(10α+1)+2048Γ(11α+1)81749606400Γ(11α+1)t11α+]x2,v3(x,t)=[tα16t3α+1120t5α15040t7α+1362880t9α139916800t11α+16227020800t13α11307674368000t15α+1355687428096000t17α+1355687428096000t17α1121645100408832000t19α+]x2.

For special case α = 1, the solution will be as follows:

v1(x,t)=(1t+12t216t3+124t41120t5+1720t615040t7+140320t81362880t9+13628800t10139916800t11+)x2=x2et,v2(x,t)=(112t+18t2148t3+1384t413840t5+146080t61645120t7+110321920t81185794560t9+13715891200t10181749606400t11+)x2=x2et2,v3(x,t)=(t16t3+1120t515040t7+1362880t9139916800t11+16227020800t1311307674368000t15+1355687428096000t171121645100408832000t19+)x2=x2sin(t).
Which is the exact solution. v1(x, t), v2(x, t) and v3(x, t) are calculated for different values of α Numerical comparisons are given in Tables 2 , 3 , 4 . It is obvious that this is a numerical solution in Fig. 1 , we plot the numerical solutions given in Eq. (12) for α = 0.5, α = 0.75 and α = 1.

Table 2

Numerical solution of v1(x, t)

xtv1FDTMfor α = 0.5v1FDTMfor α = 0.75v1FDTMfor α = 1v1Exact
0.010.010.000089597199410.000096629114150.00009900498330.00009900498337
0.020.020.00034310683240.00037763801660.00039207946940.0003920794693
0.030.030.00074723324260.00083263262730.00087340098020.0008734009802
0.040.040.0012931798640.0014530537080.0015372631030.001537263103
0.050.050.0019742251810.0022314084420.0023780735610.002378073561
0.060.060.0027849189050.0031609629630.0033903523210.003390352321
0.070.070.0037206867200.0042355723710.0045687297180.004568729718
0.080.080.0047776006410.0054495723030.0059079446180.005907944617
0.090.090.0059522316510.0067977035730.0074028426010.007402842601
0.10.10.0072415485600.0082750566510.0090483741810.009048374180
Table 3

Numerical solution of v2(x, t)

xtv2FDTMfor α = 0.5v2FDTMfor α = 0.75v2FDTMfor α = 1v2Exact
0.010.010.00009583789020.00009857498860.000099501247920.0000995012479
0.020.020.00037683059370.00039048806490.00039601993350.0003960199335
0.030.030.00083685434040.00087118442600.00088660074560.0008866007456
0.040.040.0014714633030.0015368151440.0015683178770.001568317877
0.050.050.0019742251810.0022314084420.0023780735610.002378073561
0.060.060.0032501220110.0034093380650.0034936039210.003493603921
0.070.070.0043879939610.0046100740860.0047314665400.004731466540
0.080.080.0056878625250.0059833223780.0061490524100.006149052411
0.090.090.0071471722120.0075264042320.0044435796030.004443579603
0.10.10.0087634945800.0092367530300.00954122942450.0095412294245
Table 4

Numerical solution of v3(x, t)

xtv3FDTMfor α = 0.5v3FDTMfor α = 0.75v3FDTMfor α = 1v3Exact
0.010.010.00000998334160.000003161750649.99983333 · 10−79.999833334 · 10−7
0.020.020.00005638016910.000021263156730.000007999466680.00000799946667
0.030.030.00015510631820.000064819738590.000026995950180.00002699595018
0.040.040.00031787092940.00014291761570.000063982934690.00006398293470
0.050.050.00055437015180.00026385051720.00012494792320.0001249479232
0.060.060.00087302456140.00043536310020.00021587042330.0002158704233
0.070.070.0012813461130.00066478045200.00034271995200.0003427199520
0.080.080.0017861538080.00095908787390.00051145404150.0005114540414
0.090.090.0023937136740.0013249845490.00072801624850.0007280162485
0.10.10.0031098359290.0017689218630.00099833416640.0009983341665
Fig. 1
Fig. 1

Exact solution of v1(x, t).

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 2
Fig. 2

Values of v1(x, t) for α = 1.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 3
Fig. 3

Values of v1(x, t) for α = 0.5.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 4
Fig. 4

Values of v1(x, t) for α = 0.75.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 5
Fig. 5

Exact solution of v2(x, t).

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 6
Fig. 6

Values of v2(x, t) for α = 1.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 7
Fig. 7

Values of v2(x, t) for α = 0.5

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 8
Fig. 8

Values of v2(x, t) for α = 0.75.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 9
Fig. 9

Exact solution of v3(x, t).

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 10
Fig. 10

Values of v3(x, t) for α = 1.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 11
Fig. 11

Values of v3(x, t) for α = 0.5.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

Fig. 12
Fig. 12

Values of v3(x, t) for α = 0.75.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00011

5 Conclusions

The generalized differential transformation method displayed in this work is an effective method for the numerical solution of a fractional partial differential-algebraic equation system. With full solutions, approximate solutions collected by the GDTM were compared to shapes and charts. On the other hand, the results are quite reliable for solving this problem. The exact closed-form solution was obtained for all the examples presented in this paper. FDTM offers an excellent opportunity for future research. As a result of this comparison, it is seen that the solutions obtained by the generalized differential transformation method are harmonious with the full solutions.

References

  • [1]

    A. Arikoglu and I. Ozkol, 1. (2007), Solution of fractional differential equations by using differential transform method, Chaos, Solitons and Fractals, 34, 1473–1481.

  • [2]

    G. Dilek. Kk, M. Yigider and E. Çelik, 2. (2014), Numerical Solution of Fractional Partial Differential-Algebraic Equation by Adomian Decomposition Method and Multivariate Pade Approximation, British Journal of Applied Science & Technology, 4(25), 3653–3664.

  • [3]

    MJ. Jang, CL. Chen, YC. Liu, 3. (2001), Two- dimensional differential transform for Partial differential equations, Appl. Math. Comput., 121, 261–270.

  • [4]

    F. Kangalgil, F. Ayaz, 4. (2009), Solitary wave solutions for the KdV and mKdV equations by differential transform method Chaos Solitons Fractals, 41(1), 464–472.

  • [5]

    KSV. Ravi, K. Aruna, 5. (2009), Two-dimensional differential transform method for solving linear and non-linear Schrdinger equations, Chaos Solitons Fractals, 41(5), 2277–2281.

  • [6]

    K. Tabatabaei, E. Çelik, and R. Tabatabaei, 6. (2012), Solving heat-like and wave-like equations by the differential transform method, Turk J Phys, 36, 87–98.

  • [7]

    Hatıra Günerhan, 7. (2019), Numerical Method for the Solution of Logistic Differential Equations of Fractional Order, Turkish Journal of Analysis and Number Theory, 7(2), 33–36.

  • [8]

    F. Ayaz, 8. (2004), Solutions of the system of differential equations by differential transform method. Appl. Math. Comput. 147, 547–567.

  • [9]

    F. Ayaz, 9. (2004), Applications of differential transform method to differential-algebraic equations, Appl. Math. Comput., 152, 649–657.

  • [10]

    I.H. Abdel-Halim Hassan, 10. (2008), Application to differential transformation method for solving systems of differential equations, Appl. Math. Modell. 32(12), 2552–2559.

  • [11]

    A. Yoku S. Glbahar, 11. (2019), Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation, Applied Mathematics and Nonlinear Sciences, 4(1), 35–42.

  • [12]

    D. W. Brzeziski, 12. (2018), Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus,Applied Mathematics and Nonlinear Sciences,3(2), 487–502.

  • [13]

    D. W. Brzeziski, 13. (2017), Comparison of Fractional Order Derivatives Computational Accuracy - Right Hand vs Left Hand Definition, Applied Mathematics and Nonlinear Sciences, 2(1), 237–248.

  • [14]

    I. K., Youssef, & M. H. El Dewaik, 14. (2017), Solving Poisson’s Equations with fractional order using Haarwavelet, Applied Mathematics and Nonlinear Sciences, 2(1), 271–284.

  • [15]

    M.T. Genoglu, H.M. Baskonus, & H. Bulut, 15. (2017), Numerical simulations to the nonlinear model of interpersonal relationships with time fractional derivative, AIP Conf. Proc., 1798, (020103), 1–9.

  • [16]

    D. Kumar, J. Singh, H. M. Baskonus, et al., 16. (2017), An Effective Computational Approach for Solving Local Fractional Telegraph Equations, Nonlinear Science Letters A: Mathematics, Physics and Mechanics, 8(2), 200–206.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    A. Arikoglu and I. Ozkol, 1. (2007), Solution of fractional differential equations by using differential transform method, Chaos, Solitons and Fractals, 34, 1473–1481.

  • [2]

    G. Dilek. Kk, M. Yigider and E. Çelik, 2. (2014), Numerical Solution of Fractional Partial Differential-Algebraic Equation by Adomian Decomposition Method and Multivariate Pade Approximation, British Journal of Applied Science & Technology, 4(25), 3653–3664.

  • [3]

    MJ. Jang, CL. Chen, YC. Liu, 3. (2001), Two- dimensional differential transform for Partial differential equations, Appl. Math. Comput., 121, 261–270.

  • [4]

    F. Kangalgil, F. Ayaz, 4. (2009), Solitary wave solutions for the KdV and mKdV equations by differential transform method Chaos Solitons Fractals, 41(1), 464–472.

  • [5]

    KSV. Ravi, K. Aruna, 5. (2009), Two-dimensional differential transform method for solving linear and non-linear Schrdinger equations, Chaos Solitons Fractals, 41(5), 2277–2281.

  • [6]

    K. Tabatabaei, E. Çelik, and R. Tabatabaei, 6. (2012), Solving heat-like and wave-like equations by the differential transform method, Turk J Phys, 36, 87–98.

  • [7]

    Hatıra Günerhan, 7. (2019), Numerical Method for the Solution of Logistic Differential Equations of Fractional Order, Turkish Journal of Analysis and Number Theory, 7(2), 33–36.

  • [8]

    F. Ayaz, 8. (2004), Solutions of the system of differential equations by differential transform method. Appl. Math. Comput. 147, 547–567.

  • [9]

    F. Ayaz, 9. (2004), Applications of differential transform method to differential-algebraic equations, Appl. Math. Comput., 152, 649–657.

  • [10]

    I.H. Abdel-Halim Hassan, 10. (2008), Application to differential transformation method for solving systems of differential equations, Appl. Math. Modell. 32(12), 2552–2559.

  • [11]

    A. Yoku S. Glbahar, 11. (2019), Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation, Applied Mathematics and Nonlinear Sciences, 4(1), 35–42.

  • [12]

    D. W. Brzeziski, 12. (2018), Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus,Applied Mathematics and Nonlinear Sciences,3(2), 487–502.

  • [13]

    D. W. Brzeziski, 13. (2017), Comparison of Fractional Order Derivatives Computational Accuracy - Right Hand vs Left Hand Definition, Applied Mathematics and Nonlinear Sciences, 2(1), 237–248.

  • [14]

    I. K., Youssef, & M. H. El Dewaik, 14. (2017), Solving Poisson’s Equations with fractional order using Haarwavelet, Applied Mathematics and Nonlinear Sciences, 2(1), 271–284.

  • [15]

    M.T. Genoglu, H.M. Baskonus, & H. Bulut, 15. (2017), Numerical simulations to the nonlinear model of interpersonal relationships with time fractional derivative, AIP Conf. Proc., 1798, (020103), 1–9.

  • [16]

    D. Kumar, J. Singh, H. M. Baskonus, et al., 16. (2017), An Effective Computational Approach for Solving Local Fractional Telegraph Equations, Nonlinear Science Letters A: Mathematics, Physics and Mechanics, 8(2), 200–206.

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