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Study of Nonlinear Fuzzy Integro-differential Equations Using Mathematical Methods and Applications

Jamshad Ahmad1, Angbeen Iqbal1, and Qazi Mahmood Ul Hassan2

1Department of Mathematics, Faculty of Science, University of Gujrat, Punjab, Pakistan
2Department of Mathematics, University of Wah, Punjab, Pakistan
Received December 11, 2020; Revised February 1, 2021; Accepted February 22, 2021.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this study, the homotopy perturbation sumudu transform method (HPSTM) is employed to find the analytical fuzzy solution of nonlinear fuzzy integro-differential equations (FIDEs). The solutions of FIDEs are more generalized and have better applications. The fuzzy concept is used to overrule the uncertainty in physical models. Based on the parametric form of the fuzzy number, the nonlinear integro-differential equation (IDE) is converted into two systems of nonlinear IDEs of the second kind. Some numerical examples were solved to demonstrate the efficiency and capability of the method. Graphical representations reveal the symmetry between lower and upper cut representations of fuzzy solutions and may be helpful for a better understanding of fuzzy control models, artificial intelligence, medical science, quantum optics, measure theory, and so on.
Keywords : Sumudu transform, Homotopy perturbation method, Non linear fuzzy integro-differential equation, Fuzzy solution
1. Introduction

2. Preliminaries

### Definition 2.1

The Sumudu transform of any set B is defined as

$B={v:v(t)0)}.$

The Sumudu transform method [18] is written as follows:

$S (f(t))=∫0∞f (ut) e-t=F (u), uɛ (-k1,k2).$

### Definition 2.2

The general form of the FIDE is as follows:

$y(m)(t,β)=g(t,β)+λ∫a(t)b(t)k(t,s)F (y(s,β)) ds.$

y(0, β) = (a0, b0), y′(0, β) = (a1, b1), y″(0, β) = (a2, b2), ..., ym−1(0, β) = (am−1, bm−1), where (aj, bj) with j = 0, 1, 2, 3, ...m − 1 are known to remain constant. ym(t, β) is the mth order derivative of the fuzzy function and is already given, β is a fuzzy parameter with values between [0, 1](0 ≤ β ≤ 1), λ is known to be a constant parameter, and K(t, s) is the kernel of this FIDE and depends on variable t and s. a(t) and b(t) are known to be the limits of this FIDE. If these limits are constant, the IDE will be called a fuzzy-Fredholm IDE, and if one of these limits is variable, then this equation will be said fuzzy-Volterra IDE:

In parametric form Eq. (3) is represented as

${y_(m)(t,β)=g_(t,β)+λ∫a(t)b(t)k(t,s)F (y(s,β))_ds,y¯(m)(t,β)=g¯(t,β)+λ∫a(t)b(t)k(t,s)F (y(s,β))¯ds.$

In addition, 0≤β≤1 and F(y(s, β))=(F(y(s, β)), (y(s, β))), g(t, β) = (g(t, β), (t, β)), with kernel

${K(t,s)F (y(s,β))_=K(t,s)F_(y(s,β)), (t,s)≥0,K(t,s)F (y(s,β))¯=K(t,s)F¯(y(s,β)),K(t,s)≤0.$
3. Methodology Description

To illustrate the basic representation of the HPSTM on the FIDE in general form, the basic fuzzy condition is defined in Eq. (3)

${H(x_,p,β)=(1-p) [x_(m)(t,β)-y_0(t,β)]+p [x_(m)(t,β)-g_(t,β)-∫a(t)b(t)k(t,s)F_(x(s,β))ds]=0,H(x¯,p,β)=(1-p) [x¯(m)(t,β)-y¯o(t,β)]+p [x¯(m)(t,β)-g¯(t,β)-∫a(t)b(t)k(t,s)F¯(x(s,β))]=0.$

And the initial approximation is reserved as

${x_0(t,β)=x_(t,β),x¯0(t,β)=g¯(t,β).$

Substituting Eq.(4) in Eq.(3) as

${x_(m)(t,β)=g_(t,β)+p∫a(t)b(t)k(t,s)F_ (x(s,β)) ds,x¯(m)(t,β)=g¯(t,β)+p∫a(t)b(t)k(t,s)F¯ (x(s,β)) ds.$

Applying Sumudu transform on both sides of Eq. (7) as

${S {x_(m)(t,β)}=S {g_(t,β)+p∫a(t)b(t)k(t,s)F_(x(s,β)) ds},S {x¯(m)(t,β)}=S {g¯(t,β)+p∫a(t)b(t)k(u,v)F¯(x(s,β)) ds}.$

Applying the differential property of Sumudu transform on both sides of Eq. (8).

${S {x_(t,β)}=(vm) ({x_(0,β)+x_′(0,β)vm-1+x_′(0,β)vm-2+⋯+x_m-1(0,β)v}+S{g_(t,β)+p∫a(t)b(t)k(t,s)F_(x(s,β)) ds}),S {x¯(t,β)}=(vm) ({x_(0,β)+x_′(0,β)vm-1+x_′(0,β)vm-2+⋯+x¯m-1(0,β)v}+S{g_(t,β)+p∫a(t)b(t)k(t,s)F¯(x(s,β)) ds}).$

The inverse transform on each side of Eq. (9) gives

${x_(t,β)=S-1{(vm) ({x_(0,β)+x_′(0,β)vm-1+x_′(0,β)vm-2+⋯+x_m-1(0,β)v}+S {g_(t,β)+p∫a(t)b(t)k(t,s)F_(x(s,β)) ds})},x¯(t,β)=S-1{(vm) ({x¯(0,β)+x_′(0,β)vm-1+x_′(0,β)vm-2+⋯+x¯m-1(0,β)v},+S {g¯(t,β)+p∫a(t)b(t)k(t,s)F¯(x(s,β)) ds}.)}.$

Assuming the solution of Eq. (10) is stated in power series of p

${x_(t,β)=∑j=0∞pjx_j,x¯(t,β)=∑j=0∞pjx¯j.$

By replacing the solution of Eq. (11) in Eq. (10) and comparing the coefficients as power of p, we obtain the iterations as follows:

$p0={x_0(t,β)=S-1{vm ({x_(0,β)+x_′(0,β)vm-1+x_″(0,β)vm-2+⋯+x_m-1(0,β)v}+S {g_(t,β)})},x¯0(t,β)=S-1{vm ({x¯(0,β)+x¯′(0,β)vm-2+x¯″(0,β)vm-3+⋯+x¯m-1(0,β)},S {g¯(t,β)})},$$p1={x_1(t,β)=S-1{vm (S{p∫a(t)b(t)k(t,s)F_(x0(s,β))ds})},x¯1(t,β)=S-1{vm (S{p∫a(t)b(t)k(t,s)F¯(x0(s,β))ds})},$$p2={x_2(t,β)=S-1{(vm) (S{p∫a(t)b(t)k(t,s)F_(x1(s,β))ds})},x¯2(t,β)=S-1{(vm) (S{p∫a(t)b(t)k(t,s)F¯_(x1(s,β))ds})},$

and so on. Finally, the solution of FIDE-2 is given as

${y_(t,β)=limp→1x_(t,β)=∑j=0∞x_j(t,β),y¯(t,β)=limp→1 x¯(t,β)=∑j=0∞x¯j(t,β).$
4. Numerical Applications

### Problem 4.1

Consider the nonlinear fuzzy Fredholm IDE of the second kind as [19]

$y′(t,β)=g(t,β)+∫01t2s10y2(s,β)ds,$

with initial condition y(0, β) = (0, 0), λ = 1, 0 ≤ s ≤ 1, 0 ≤ β ≤ 1, $K(t,s)=t2s10,g(t,β)=(β-t2β240,(2-β)-t2(2-β)240)$.

The exact solution of Eq. (16) is given as y(t, β) = (, (2 − β)).

The homotopy is

${H(x_,p,β)=x_′(t,β)-(β-t2β240)-p∫01t2s10x_2(s,β)ds=0,H(x¯,p,β)=x¯′(t,β)-((2-β)-t2(2-β)240)-p∫01t2s10x¯2(s,β)ds=0.$

According to the procedure described above, we have

${x_(t,β)=S-1{v {S{(β-t2β240)+p∫01t2s10x_2(s,β)ds}}},x¯(t,β)=S-1{v {S{((2-β)-t2(2-β2)40)+p∫01t2s10x¯2(s,β)ds}}}.$

Consequently, we have

$p0:{x_0(t,β)=tβ-t3β2120,x¯0(t,β)=t(2-β)-t3(2-β2)120,$$p1:{x_1(t,β)=t33!(β220+β4576000-β33600),x¯1(t,β)=t33!((2-β)220+(2-β)4576000-(2-β)33600),$

and so on. The series solution (Figure 1) is

${y_(t,β)=tβ-t3β2120+t3β2120+t3β4345600-t3β321600+⋯,y¯(t,β)=t(2-β)-t3(2-β)2120+t3(2-β)2120+t3(2-β)4345600-t3(2-β)321600+⋯.$

### Problem 4.2

Consider the nonlinear fuzzy Volterra IDE of the second kind as [9]

$y′(t,β)=(β,(7-β))+∫0ty2(s,β)ds,$

where y(0, β) = (0, 0), 0 ≤ s ≤ 1, 0 ≤ α ≤ 1, λ = 1, K(t, s) = 1, g(t, β) = (β, (7 − β)). According to the procedure described above, we have

${x_(t,β)=S-1 {v {S {β+p∫0tx_2(s,β)ds}}},x¯(t,β)=S-1 {v {S {(7-β)+p∫0tx¯2(s,β)ds}}},$$p0:{x_0(t,β)=tβ,x¯0(t,β)=t(7-β),$$p1:{x_1(t,β)=β2t412,x¯1(t,β)=(7-β)2t412,$$p2:{x_2(t,β)=β4t1012960,x¯2(t,β)=(7-β)4t1012960,$

and so on. The solution (Figure 2) is given as

${y_(t,β)=tβ+β2t412+β4t1012960+⋯,y¯(t,β)=t(7-β)+(7-β)2t412+(7-β)4t1012960+⋯.$

### Problem 4.3

Consider the nonlinear fuzzy Volterra IDE of 2nd second-order as [11]

$y″(t,β)=g(t,β)+∫0ty2(s,β)ds,$

with initial values y(0, β) = (β − 1, 1 − β), y′(0, β) = (β, 2 − β), where g(t, β) = (−β2t3, −(2 − β)2t3), 0 ≤ st, 0 ≤ βt, λ = 1, K(t, s) = 1.

The exact solution is given by y(0, β) = (3, 3t (2 − β)) . The homotopy as

${H(x_,p,β)=x_″(t,β)+β2t3-p∫0tx_2(s,β)ds=0,H(x¯,p,β)=x¯″(t,β)+(2-β)2t3-p∫0tx¯2(s,β)ds=0,$

According to the procedure described above, we have

${x_(t,β)=S-1 {v2 (β-1v2+βv+S {-β2t3+p∫0tx_2(s,β)ds})},x_(t,β)=S-1{v2 (1-βv2+2-βv+S {-(2-β)2t3+p∫01x¯2(s,β)ds})}.$

The solution (Figure 3) in powers of p is represented as

$p0:{x_0(t,β)=(β-1)+βt-β2t520β2x¯0(t,β)=(1-β)+t(2-β)-(2-β)2t520,$$p1:{x_1(t,β)=t36(β-1)+t412β(β-1)+t560β2+t13686400β4-t93024β2-t83360β2(β-1),x_1(t,β)=t36(1-β)+t412(2-β)(1-β)+t560(2-β)2+t13686400(2-β)4-t93024(2-β)2-t83360(2-β)2(1-β),y_(t,β)=t(β-1)+t22β-t520β2+t5120(β-1)2-t7840β2+t131716β4+t7840β(β-1)$${-t1014400β3-t910080β2(β-1),y¯(t,β)=t(1-β)+t22(2-β)-t520(2-β)2+t5120(1-β)2+t7840(2-β)2+t131716(2-β)4+t7840(2-β)(1-β)-t1014400(2-β)3-t910080(2-β)2(1-β).$

### Problem 4.4

Consider the nonlinear IDE of the second kind [11]

$y′(t,β)=g(t,β)+∫01s22y2(s,β)ds,$

with the initial condition y(0, β) = (0, 0), where 0 ≤ t, s ≤ 1, 0 ≤ β ≤ 1, λ = 1, $K(t,s)=s22.g(t,β)=((β-β28),(12-4β-β2))$, The exact solution is (βt, (2 − β) t) . According to the detail procedure described above, we have

${x_(t,β)=S {v (S {(β-β28)+p∫01s22x_2(s,β)ds})},x¯(t,β)=S {v (S {(β-β28)+p∫01s22x¯2(s,β)ds})}.$

The solution components are

$p0:{x_0(t,β)=-t8(-8+β)β,x0¯(t,β)=-t8(-12+4β+β2),$$p1:{x_1(t,β)=t640(-8+β)2β2,x1¯(t,β)=t640(-12+4β+β2)2,$$p2:{x_2(t,β)=t4096000(-8+β)4β4,x2¯(t,β)=t4096000(-12+4β+β2)4,$

and so on. The solution (Figure 4) is represented by utilizing the above iterations as

${y_(t,β)=-t8(-8-β)β+t640(-8-β)2β2+t4096000(-8-β)4β4y¯(t,β)=-t8(-12+4β+β2)+t640(-12+4β+β2)2t4096000(-12+4β+β2)4,$

### Problem 4.5

Finally, consider the nonlinear fuzzy VIDE of the first order as [11]

$y′(t,β)=g(t,β)+∫0ty2(s,β)ds,$

with initial condition y(0, β) = (0, 0), where 0 ≤ t, s ≤ 1, 0 ≤ β ≤ 1, λ = 1, K(t, s) = 1, and

$g(t,β)=((-15 (β4+2β3+β2) t5+2 (β2+β) t),(-15 (16-8β3-8β+β6+2β4+β2) t5+2 (4-β3+β) t),).$

The exact solution of Eq. (40) is given as

$y(t,β)=((β2+β)t2,(4-β3+β)t2).$

We construct the homotopy as

${H(x,p,β)=x′_(t,β)+15 (β4+2β3+β2) t5-2(β2+β)t-p∫0tx_2(s,β)ds=0,H(x¯,p,β)=x′¯(t,β)+15(16-8β3-8β+β6+2β4+β2)t5-2 (4-β3+β) t-p∫0tx¯2(s,β)ds=0.$

According to the procedure described above, we have

${x_(t,β)=S-1 {v (S{-15 (β4+2β3+β2) t5+2 (β2+β) t+p∫0tx_2(s,β)ds})},x¯(t,β)=S-1 {v (S{-15 (16-8β3-8β+β6+2β4+β2) t5+2 (4-β3+β) t+p∫0tx¯2(s,β)ds})}$

Comparing the powers of parameter p in Eq. (42), we have

$p0:{x_0(t,β)=(β2+β) t2-130 (β4+2β3+β2) t6,x0¯(t,β)=(4-β3+β) t2-130(16-8β3-8β+β6+2β4+β2)t6,$$p1:{x_1(t,β)=(β2+β)2t630+(β4+2β3+β2)2t14163800,-(β2+β) (β4+2β3+β2)t101350,x1¯(t,β)=(4-β3+β)2t630+(16-8β3-8β+β6+2β4+β2)2t14163800,-(4-β3+β) (16-8β3-8β+β6+2β4+β2)t101350,$

and so on. Utilizing the above results and we get series solution (Figure 5) as

${y_(t,β)=(β2+β) t2-130 (β4+2β3+β2) t6+130(β2+β)2t6+(β4+2β3+β2)2t14163800-(β2+β) (β4+2β3+β2)t101350,y¯(t,β)=(4-β3+β)t2-130(16-8β3-8β+β6+2β4+β2) t6+(4-β3+β)2t630+(16-8β3-8β+β6+2β4+β2)2163800-(4-β3+β) (16-8β3-8β+β6+2β4+β2)t101350.$
5. Conclusion

In this study, we investigated the HPSTM in order to solve non-linear FIDEs based on the parametric form of fuzzy numbers. The solutions obtained by these fuzzy differential equations are considered to be controllers in applications. This method proved its effectiveness and reliability in solving uncertain types of equations by providing the best approximate solutions. The numerical outcomes obtained using the proposed technique are comparable to the exact solutions. Thus, the work can be extended to higher-order mixed-type FIDEs in more variables.

Conflict of Interest

The authors have no conflict of interest.

Figures
Fig. 1.

Comparison of approximate solution (21) and exact solution of (a) y and (b) at t = 1.

Fig. 2.

Comparison of approximate solution (27) and exact solution of y and (a) at (b) t = 1.

Fig. 3.

Comparison of approximate solution (33) and exact solution of (a) y and (b) at t = 1.

Fig. 4.

Comparison of approximate solution (39) and exact solution of (a) y and (b) at t = 1.

Fig. 5.

Comparison of approximate solution (45) and exact solution of (a) y and (b) at t = 1.

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Biographies

Jamshad Ahmad is currently working as an Assistant Professor in the Department of Mathematics, University of Gujrat, Pakistan. He received his Ph.D. degree in 2015 from HITEC University, Taxila, Pakistan. His research interests include computational and applied mathematics.

E-mail:

Angbeen Iqbal received his Master of Philosophy in Mathematics in 2019 from the University of Gujrat, Pakistan. She did her Bachelor of Science degree in Mathematics in 2017 from the University of Gujrat, Pakistan. Her research interests include fuzzy differential equations and fuzzy fractional differential equations.

E-mail:

Qazi Mahmood Ul-Hassan is currently working as an Associate Professor in the Department of Mathematics, University of Wah, Wah Cantt. Pakistan. He received his Ph.D. from HITEC University, Taxila Cantt, Pakistan. His research interests include computational and applied mathematics.

E-mail:

March 2021, 21 (1)