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Математика
DOI: 10.14529/mmph210301
ON THE EXACT SOLUTIONS TO CONFORMABLE EQUAL WIDTH WAVE EQUATION BY IMPROVED BERNOULLI SUB-EQUATION FUNCTION METHOD
V. Ala, U. Demirbilek, K.R. Mamedov
Mersin University, Mersin, Turkey E-mail: [email protected]
In this paper, we consider conformable equal width wave (EW) equation in order to construct its exact solutions. This equation plays an important role in physics and gives an interesting model to define change waves with weak nonline-arity. The aim of this paper is to present new exact solutions to conformable EW equation. For this purpose, we use an effective method called Improved Bernoulli Sub-Equation Function Method (IBSEFM). Based on the values of the solutions, the 2D and 3D graphs and contour surfaces are plotted with the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable partial equations arising in mathematical physics.
Keywords: improved Bernoulli sub-equation function method; conformable equal width wave equation; wave transformation.
Introduction
Fractional differential equations are the generalization of classical differential equations with integer order. So, in recent years, fractional differential equations become the field of physicists and mathematicians who investigate the expediency of such non-integer order derivatives in different areas of physics and mathematics. These equations became a useful tool for describing numerous nonlinear phenomena of physics such as heat conduction systems, nonlinear chaotic systems, viscoelasticity, plasma waves, acustic gravity waves, diffusion processes [1-3]. Many numerical and analytical methods were developed and successfully employed to solve these equations such as modified Kudryashov method [4], homotopy perturbation method [5], new extended direct algebraic method [6], fractional Riccati expansion method [7], modified extended tanh method [8].
During the last few years, a straightforward definition of conformable derivative is given [9]. The conformable derivative operator is compatible to many real-world problems and provides some properties of classical calculus such as derivative of quotient or product of two functions, the chain rule [10]. During the last few years, many of techniques applied to find exact solutions to conformable nonlinear partial differential equations [11-16].
In this paper, we consider the following conformable EW equation:
Dfu + pD«u2-iD^u = 0, a e (0,1], (1)
where p,i are real parameters, u is a function of independent variables. The operator Da represents conformal derivative operator defined only for positive region of t [10]. Before consideration of the solution procedure, let us give some properties of conformable derivative.
The conformable derivative of order a with respect to the independent variable t is defined as [9]
Da(y(t)) = lim y(t + r t1-a) -y(t),t > 0,a e (0,1]
for a function y = y(t): [0, oo) —» M.
Theorem 1.1. Assume that the order of the derivative is a e (0,1] and suppose that u = u(t) and v = v(t) are a - differentiable functions for all positive t. Then i. D"(cxu + c2v) = cxD*(u) + c2D"(v) for V cx,c2 el,
ii. Df{tk) = ktk~a V keR,
iii. D^ (A) = 0 for each constant function u (t) = A,
iv. Da (uv) = uD^ (v) + vD^ (u),
.fl| u vD (u) -uDa(v)
Da - =
? v) v2
vi. Da (u) (t )=t1-« .
Conformable differential operator satisfies some critical fundamental properties like the chain rule, Taylor series expansion and Laplace transform.
Theorem 1.2. Let u = u(t) be an a-conformable differentiable function and assume that v is a differentiable function. Then
Da (u o v)(0 = tl~av (t)u (v(0). The proofs of Theorems 1 and 2 are given in [17] and [9], respectively.
The rest part of the paper is organized as follows. In Section 1, descripton of the IBSEFM is given. In Section 2, the application of IBSEFM is mentioned. Finally, this study is completed by providing conclusions in the last section.
1. Description of the IBSEFM
In this section, we give the fundamental properties of the IBSEFM. This method is direct, significant, advanced algebraic method to establish reliable exact solutions for both nonlinear and nonlinear fractional partial differential equations [11,18-20]. Let us describe five main steps of the IBSEFM.
Step 1. Let us take into account the following conformable partial differantial equation of the form
P(v, D(a)v, D(xa)v, Dfta)v,...) = 0, (2)
where Dta) is the conformable fractional derivate operator, v(x, t) is an unknown function, P is a polynomial and its partial derivatives contain fractional derivatives. The aim is to convert conformable nonlinear partial differential equation with a suitable fractional transformation into the ordinary differantial equation. The wave transformation is
v(x,t) = V(4), 4 = 4(x,ta). (3)
Using the properties of conformable fractional derivate, we convert (2) into an ODE of the form
N(V,V' ,V" ,...) = 0. (4)
If we integrate (4) term to term, we acquire integration constant(s), which can be determined later. Step 2. We hypothesize that the solution to (4) can be represented as follows:
Z aF (4)
v(4) = io__ao + atF (4) + at F2 (4) +... + a„Fn (4) (5)
f^b .FJ (4) bo + bjF(4) + btF2(4) +... + bmFm(4)
j=o
where a0,ax,...,an and b0,bj,...,bm are coefficients which will be determined later. The numbers m ^ 0, n ^ 0 are chosen arbitrary constants to balance principle. Consider the Bernoulli differential equation of the form
F (4) = <tF(4) + dFM (4), d * 0, a * 0,M e W {0,1,2}, (6)
where F (4) is the solution to (6).
Step 3. The positive nonzero integers m, n,M are found by balance principle that is both nonlinear term and the highest order derivative term of (4).
Substituting (5), (6) in (2) gives the following equation of the polynomial ©(F) of F :
_©(F(4)) = psF(4)s +... + px F (4) + p0 =0,_
g Bulletin of the South Ural State University
Ser. Mathematics. Mechanics. Physics, 2021, vol. 13, no. 3, pp. 5-13
On the Exact Solutions to Conformable Equal width Wave Equation by Improved Bernoulli Sub-Equation Function Method
where pi, i = 0,..., 5 are coefficients and will be determined later.
Step 4. Equating all the coefficients of 0(F(£)) yields us an algebraic equation system
Pi =0, i = 0,.., 5.
Step 5. When we solve (4), we get the following two cases with respect to a and d :
F (Л =
- dea(s—1) +
sa
ae
a(s-i)|
i
1—s
F Л) =
(s — 1) + (s + 1)tanh I u(1 — s)
Л
Л
1 — tanh I u(1 — s)
= cr,£ e
(7)
(8)
Using a complete discrimination system of F(£), we obtain the analytical solutions to (4) via mathematics software and categorize the exact solutions to (4). To achieve better results, we can plot two and three dimensional figures of analytical solutions by considering proper values of parameters.
2. Application of the IBSEFM
Take the travelling wave transformation
xa ta
i(x, t) = U (л), л= q--+ m—
a a
(9)
where q and m are nonzero constants to be determined. Then Equation (1) turns into the following ordinary differential equation:
mU + pqU2 -qml2U" =0. (10)
When we apply the balance principle for the terms u2 and u , the result for m, n and M is
M + m = n +1.
By balancing the order between the nonlinear term and highest order derivative in Equation (10), we obtain M = 3, n = 3 and m = 1, then we get
Ufa) = ao + aiFM + a2F2 M + a3F3 M) _ Tfa)
bo + byF (л) , T (л)У(л) — T(])¥ (л) 1 ^2fa) :
w (л)'
T (л)w(л) — Т(л)w (л) Т(л)w (л) 'w2 (л) — 2T(])[W' (л)]2wл) U (л) =-—------
w2 (л)
w4 (л)
Substituting (3)—(11) in (10), we get the algeabraic equation system according to F : F0 :2pqa^b0 + maob0 = 0,
F : 4pqa0a1b0 + mab — lmq2a2a1b02 + 2pqa^b1 + 2ma0b0b1 + lmqza2a0b0b1 = 0
2_2,
F : 2pqa1 b0 + 4pqa0a2b0 + ma2b0 — 4lmq a a2 b0 + 4pqa0a1b1 + 2ma1b0b1 + Imq a ap0\ + ma0b2 — lmq2a2a0b2 = 0,
F :4pqaflb + 4pqaoba — 4dlmq aab + ma3b0 — 9lmq a a3b0 + 2pqaxbx + 4pqa0a2b + +4dlmq2cra(jb(jbl + 2ma2h{)h} - ?>lmq2cj2a2bQbl + maxb2 = 0,
(11) (12)
Q О О О О О О
F : 2pqa4b0 + 4pqa3a5b0 — 24d lmq a4b0 + 4pqa3a4b1 + 4pqa2a5b1 — 21d lmq a3b0b1 —
О ООО 00
—96dlmq aa5b0b1 — 3d lmq a2b1 — 24dlmq aa4b1 = 0,
1 П ООО О OO ООО
F :4 pqa4a5b0 — 35d lmq a5b0 + 2 pqa4 b1 + 4 pqa3a5b1 — 37d lmq a4b0b1 — 8d lmq a3b1 —
-40dlmq2aa5b2 = 0,
112 22 222 F :2pqa5b0 + 4pqa4a5b1 - 57d Imq a5b0b1 -15d Imq a4b1 =0,
F12 : 2pqa52b1 - 24d2lmq2a5b12 = 0.
Then we solve the system of equations of F and, in each case, substitute the obtained coefficients to get the new solution(s) u(x, t ). Solving the system by Wolfram Mathematica software, we obtain the coefficients as follows. Case 1. For a ^ d,
u a
u2a
4. "5 . _^a5 _ a5bQ
a =
ua^
0 6d 2; ai 6d 2'a2
a=
d
a=
; a3
d
; bi= A m = _ pu2; ^ -_i
3d 2b
4q2u2
(13)
4 3d b0
where J, l, q,a, a4, b0 ^ 0 .
Putting (13) along with (3)-(11) in (7), we acquire the exponential function solution to the EW equation as follows:
( 1 6 6 ^
^4
u
u1( x, t) =
(
2 pu
< ai px«a aA ^
(
3ta
d 2 b
о J
V d - e
3a
su
2 pu
< a+pxOUaA^ d2 b
3ta
V
^0 J
V d - e
3a
J su J
Fig. 1. The contourplot, 3D and 2D graphs of u(x,t) by considering the values a = 0,1; s = 0,2; a = 0,1; d = 0,3; p = 0,5; q = 0,6; b0 =1; a =1; -13 < x <13, -2 < t <2 for 3D surface, 0< x <3, 0< t <3 for contourplot and
-1< x <1; t = 0,2 for 2D
Case 2. For u Ф d,
24d Iq
-; a =—
a5 _ ia4 _ -ia5 _ a5b0 _ 12d lmqb0 _ i
24d Iq
; a = -
2d*Jïq 3 2d^q
; bi ; p
;u=-
l^fïq
(14)
where d,l, q,a4,b0 ^ 0 . Substituting (14) along with (3)—(11) in (7), we obtain solution to (1)
f \
-l -
i( qta+mxa^j 12ide **fiqa -Jïqs
u2 ( x, t ) =
24d 2lq 2b0
a
On the Exact Solutions to Conformable Equal width Wave Equation by Improved Bernoulli Sub-Equation Function Method
< t » »
Fig. 2. The contourplot, 3D and 2D graphs of imaginary part of u2 (x, t) by considering the values a = 0,4; s = 0,1; u = 0,2; d = 0,51; l = 0,5; m = 0,3; q = 0,73; b0 =0,6; a4 =0,2; -17 <x<17, -7 <t <7 for 3D surface, -5 <x<5,
-5< t <5 for contourplot and -10 < x <10; t = 0,2 for 2D
Fig. 3. The contourplot, 3D and 2D graphs of real part of u2 (x, t) by considering the values a = 0,4; s = 0,1; u = 0,2; d = 0,51; l = 0,5; m = 0,3; q = 0,73; b0 =0,6; a4 =0,2; -17 <x<17, -7 <t <7 for 3D surface, -5 <x<5, -5 <t <5
for contourplot and -10 < x <10; t = 0,2 for 2D
Case 3. For u Ф d,
mbn majbn H 3v mJa4Jb0 г
___J0 . _ '»45^0 . __
a0 — ; a1 — ; a2 — 2 pq 2 pqa4
•JJyfma^yfbQ ^ _ a5b0
4P4q4^ 51 a4
(15)
, VPVa4 i
d —--— ;u —--
where l,
m,p,q,a,a4,b0 ^ 0. When we put (15) along with (3)-(11) in (7), and obtain the complex solution to (1) as follows:
3m
u3 ( x, t ) = -
2ita
^ qta+mxa)
о • a
2imx
e^lapqa4 + 4i>/3e ^la
a
ms2bn
2 pq
2imx
s[p<fqyla4 + 3e^qa *Jms*Jb0
2
Fig. 4. The contourplot, 3D and 2D graphs of imaginary part of u3(x,t) by considering the values a = 0,1; s = 0,2; j = 0,3; d = 0,62; l = 0,6; m = 0,2; q = 0,47; b0 =0,8; a4 =0,5; p = 0,3; -17 <x<17, -7 <t <7 for 3D surface, -5 < x <5, -5 < t <5 for contourplot and -10 < x <10; t = 0,3 for 2D
Fig. 5. The contourplot, 3D and 2D graphs of real part of u3 (x, t) by considering the values a = 0,1; s = 0,2; j = 0,3; d = 0,62; l = 0,6; m = 0,2; q = 0,47; b0 =0,8; a4 =0,5; p = 0,3; -17 < x <17, -7 < t <7 for 3D surface, -5 < x <5,
-5 < t <5 for contourplot and -10 < x <10; t = 0,3 for 2D
Conclusion
In this paper, the IBSEFM method is applied for the conformable EW equation. Using a wave transformation, we convert the conformable differential equation into the ordinary differential equation, which can be solved according to the IBSEFM. By means of this method, exact solutions are obtained. The contourplot, 3D and 2D surfaces of all solutions obtained by IBSEFM under the suitable values of parameters are plotted to show the main characteristic physical properties of the solutions with the help of mathematics software. According to the results, one can see that the formats of travelling wave solutions in two and three dimensional surfaces are similar to the physical meaning of results.
The solutions are also solitary wave solutions. Also, it is clear that more steps are developed and better approximations are obtained. The conclusions show that the IBSEFM is simple, effective and powerful. Thus, in mathematical physics, it is applicable to solve other conformable partial differential equations. We claim that the IBSEFM method is practically well suited, since it can be adopted to a wide range of nonlinear differential equations. Eventually, this method is influential and suitable for solving other types of nonlinear differential equations in which the balance principle is satisfied.
References
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2. Podlubny I. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Mathematics in Science and Engineering, 198, San Diego, CA: Academic Press, 1999, 340 p.
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On the Exact Solutions to Conformable Equal width Wave Equation by Improved Bernoulli Sub-Equation Function Method
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12. Ala V., Demirbilek U., Mamedov Kh.R. An Application of Improved Bernoulli Sub-Equation Function Method to the Nonlinear Conformable Time-Fractional SRLW Equation. AIMS Mathematics, 2020, Vol. 5, Iss. 4, pp. 3751-3761. DOI: 10.3934/math.2020243
13. Yel G., Sulaiman T.A., Bakonu H.M. On the Complex Solutions to the (3+1)-dimensional Conformable Fractional Modified KdV-Zakharov-Kuznetsov Equation. Modern Physics Letters B, 2020, Vol. 34, Iss. 5, Article number 2050069. DOI: 10.1142/S0217984920500694
14. Durur H., Tasbozan O., Kurt A. New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations. Applied Mathematics and Nonlinear Sciences, 2020, Vol. 5, Iss. 1, pp. 447-454. DOI: 10.2478/amns.2020.1.00042
15. Kurt A. New Analytical and Numerical Results For Fractional Bogoyavlensky-Konopelchenko Equation Arising in Fluid Dynamics. Applied Mathematics-A Journal of Chinese Universities, 2020, Vol. 35, Iss. 1, pp. 101-112. DOI: 10.1007/s11766-020-3808-9
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Received February 2, 2021
Information about the authors
Ala Volkan, Mathematics Department, Science and Letters Faculty, Mersin University, Mersin, Turkey, ORCID iD: https://orcid.org/0000-0002-8499-9979, e-mail: [email protected]
Demirbilek Ulviye, Mathematics Department, Science and Letters Faculty, Mersin University, Mersin, Turkey, ORCID iD: https://orcid.org/0000-0002-5767-1089, e-mail: [email protected]
Mamedov Khanlar Rashid, Mathematics Department, Science and Letters Faculty, Mersin University, Mersin, Turkey, ORCID iD: https://orcid.org/0000-0002-3283-9535, e-mail: [email protected]
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2021, vol. 13, no. 3, pp. 5-13
УДК 517.9 DOI: 10.14529/mmph210301
О ТОЧНОМ РЕШЕНИИ СОГЛАСОВАННОГО РАВНОМОЩНОГО ВОЛНОВОГО УРАВНЕНИЯ С ПОМОЩЬЮ УСОВЕРШЕНСТВОВАННОГО ФУНКЦИОНАЛЬНОГО МЕТОДА ПОД-УРАВНЕНИЯ БЕРНУЛЛИ
В. Длэ, Ю. Демирбилек, K.Р. Мамедов
Мерсинский университет, Мерсин, Турция E-mail: volkanala@mersin. edu.tr
В настоящей работе рассматривается согласованное равномощное волновое уравнение с целью нахождения его точного решения. Данное уравнение играет важную роль в физике и задает интересную модель определения изменяющихся волн со слабой нелинейностью. Целью работы является представление нового точного решения согласованного равномощного волнового уравнения. Для этого авторы используем эффективный метод, называемый усовершенствованным функциональным методом под-уравнения Бернулли (IBSEFM). На основе значений решений, двумерные и трехмерные графики и контурные поверхности строятся с привлечением математического программного обеспечения. Полученные результаты подтверждают, что IBSEFM является мощным математическим аппаратом для решения нелинейных согласованных уравнений в частных производных, возникающих в математической физике.
Ключевые слова: усовершенствованный функциональный метод под-уравнения Бернулли; согласованное равномощное волновое уравнение; волновое преобразование.
Литература
1. Baleanu, D. Fractional Dynamics and Control / D. Baleanu, J.A.T. Machado, A.C.J. Luo (Eds.).
- Springer-Verlag New York, 2012. - 310 p.
2. Podlubny, I. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications / I. Podlubny // Mathematics in Science and Engineering. - Vol. 198. - San Diego, CA: Academic Press, 1999. -340 p.
3. Miller, K.S. An Introduction to the Fractional Calculus and Fractional Differential Equations / K.S. Miller, B. Ross. - New York: Wiley, 1993. - 366 p.
4. Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity / H. Rezazadeh, M.S. Osman, M. Eslami et al. // Optik. - 2018. - Vol. 164. -P. 84-92.
5. Yildirim, A. Homotopy Perturbation Method for Solving the Space-Time Fractional Advection Dispersion Equation / A. Yildirim, H. Ko?ak // Advances in Water Resources. - 2009. - Vol. 32, Iss. 12.
- P.1711-1716.
6. New Exact Solutions of Nonlinear Conformable Time-Fractional Phi-4 Equation / H. Rezazadeh, H. Tariq, M. Eslami et al. // Chinese Journal of Physics. - 2018. - Vol. 56, Iss. 6. - pp. 2805-2816.
7. Abdel-Salam, A. Solution of Nonlinear Space-Time Fractional Differential Equations using the Fractional Riccati Expansion Method / A. Abdel-Salam, B.A. Emad, A.Y. Eltyabev // Mathematical Problems in Engineering. - 2013. - Vol. 2013. - P. 1-6. - Article ID 846283.
8. Shallal, M.A. Analytic Solution for the Space-Time Fractional Klein-Gordon and Coupled Conformable Boussinesq Equations / M.A. Shallal, H.N. Jabbar, K.K. Ali // Results in Physics. - 2018. -Vol. 8. - P. 372-378.
9. Abdeljawad, T. On Conformable Fractional Calculus / T. Abdeljawad // J. Comput. Appl. Math.
- 2015. - Vol. 279. - P. 57-66.
Ala V., Demirbilek U., On the Exact Solutions to Conformable Equal width Wave Equation
Mamedov K.R. by Improved Bernoulli Sub-Equation Function Method
10. Khalil, R. A new Definition of Fractional Derivative / R. Khalil, M. Horani, A. Yousef, M. Sababheh // J. Comput. Appl. Math. - 2014. - Vol. 264. - P. 65-70.
11. Baskonus, H.M. Exponential Prototype Structures for (2+1)-Dimensional Boiti-Leon-Penpinelli Systems in Mathematical Physics / H.M. Baskonus, H. Bulut // Waves in Random and Complex Media. - 2016. - Vol. 26, Iss. 2. - P. 189-196.
12. Ala, V. An Application of Improved Bernoulli Sub-Equation Function Method to the Nonlinear conformable Time-Fractional SRLW Equation / V. Ala, U. Demirbilek, Kh.R. Mamedov // AIMS Mathematics. - 2020. - Vol. 5, Iss. 4. - P. 3751-3761.
13. Yel, G. On the Complex Solutions to the (3+1)-Dimensional Conformable Fractional Modified KdV-Zakharov-Kuznetsov Equation / G. Yel, T.A. Sulaiman, H.M. Bakonu // Modern Physics Letters B. - 2020. - Vol. 34, Iss. 5. - Article number 2050069.
14. Durur, H. New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations / H. Durur, O. Tasbozan, A. Kurt // Applied Mathematics and Nonlinear Sciences. - 2020. - Vol. 5, Iss. 1. - P. 447-454.
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Поступила в редакцию 2 февраля 2021 г.
Сведения об авторах
Ала Волкан, кафедра математики, факультет естественных наук и литературы, Мерсинский университет, Мерсин, Турция, ORCID iD: https://orcid.org/0000-0002-8499-9979, e-mail: [email protected]
Демирбилек Улвиё, кафедра математики, факультет естественных наук и литературы, Мерсинский университет, Мерсин, Турция, ORCID iD: https://orcid.org/0000-0002-5767-1089, e -mail : udemirbilek@mersin .edu.tr
Мамедов Ханлар Рашид, кафедра математики, факультет естественных наук и литературы, Мерсинский университет, Мерсин, Турция, ORCID iD: https://orcid.org/0000-0002-3283-9535, e-mail: [email protected]
