SOLVING FIRST ORDER NONLINEAR FUZZY DIFFERENTIAL ... - ijpam

International Journal of Pure and Applied Mathematics Volume 118 No. 1 2018, 49-64 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v118i1.5

AP ijpam.eu

SOLVING FIRST ORDER NONLINEAR FUZZY DIFFERENTIAL EQUATIONS USING OPTIMAL HOMOTOPY ASYMPTOTIC METHOD A.F. Jameel1 § , A. Saaban2 , S.A. Altaie3 , N.R. Anakira4 , A.K. Alomari5 , N. Ahmad6 1,2,3,6 School

of Quantitative Sciences Universiti Utara Malaysia (UUM) Kedah, Sintok, 06010, MALAYSIA 3 Computer Engineering Department University of Technology Baghdad, IRAQ 4 Department of Mathematics Faculty of Science and Technology Irbid National University 2600 Irbid, JORDAN 6 Department of Mathematics Faculty of Science Yarmouk University Irbid, 211-63, JORDAN

Abstract: In this paper, we discuss the approximate solution of first order nonlinear fuzzy initial value problems (FIVP) by formulate and analyze the use of the Optimal Homotopy Asymptotic Method (OHAM). OHAM allows for the solution of the fuzzy differential equation to be calculated in the form of an infinite series in which the components can be easily computed. This method provides us with a convenient way to control the convergence of approximation series. Numerical examples using the well-known nonlinear FIVP are presented to show the capability of the this method in this regard and the results are satisfied the convex triangular fuzzy number. Received: Revised: Published:

April 21, 2017 October 11, 2017 February 16, 2018

§ Correspondence author

c 2018 Academic Publications, Ltd.

url: www.acadpubl.eu

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AMS Subject Classification: 03A72, 34A07, 14F35 Key Words:

fuzzy numbers, fuzzy differential equations, first order fuzzy initial value

problems, optimal homotopy asymptotic method

1. Introduction One of the aims for studying fuzzy sets theory is to develop the methodology of formulations and to find solutions of problems that are too complicated or illdefined to be acceptable for analysis by conventional techniques. Mathematical model in many dynamical real life problems can be represented by a system of ordinary or partial differential equations. To model a dynamical system which is inadequate of it behaviors and its nature is uncertainty, the fuzzy differential equations (FDEs) are a useful tool to be considered. Since it is widely utilized for the purpose of modeling problems in science and engineering [4, 5, 6], FDEs recently becomes a popular topic studied by many researchers. The solution of a fuzzy differential equation (FDE) which satisfies fuzzy initial or boundary value conditions occurs in many practical problems. Considering the approximate solutions of fuzzy initial or boundary value problems is becoming more important because many of them could not be solved exactly and sometimes impossible to find their analytic solutions. FDE appears when the modeling of these problems was imperfect and its nature is under uncertainty. For more explanation, when a real world problem is transformed into a deterministic boundary value problem involving ordinary differential equations we cannot usually be sure that the model is perfect. For example, the boundary conditions value may not be known exactly may contain uncertain parameters. If they are estimated through certain measurements, they are necessarily subject to errors [7].The analysis of the effect of these errors leads to the study of the qualitative behavior of the solutions of these problems. Thus, it would be natural to nonlinear fuzzy initial value problems (FIVPs). Solving problems with approximate analytical methods often helps to better understand a physical problem, and may help improve future procedures and designs used to solve these problems. The essential gain of an approximate methods is being able to solve difficult nonlinear problems, without the need to compare with the exact solution to determine the accuracy of the approximate solution. In recent years semi analytical methods such as Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) have been used to solve fuzzy problems involving ordinary

SOLVING FIRST ORDER NONLINEAR FUZZY...

51

differential equations. Approximate analytical methods such as the Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) have been used to solve fuzzy initial value problems involving ordinary differential equations. HPM was proposed and utilized to solve first order FIVPs linear and nonlinear FIVPs [8, 9]. The ADM and VIM [10, 11, 12] was employed to solve first order linear and nonlinear fuzzy initial value problems. OHAM was introduced recently by Marinca et al. [13] and applied for solving nonlinear problems without depending on the small parameter [14, 15, 16]. The OHAM has also been successfully applied on various engineering problems [17, 18, 19, 31, 32]. Mabood in [20] has provided a comparative study between OHAM and HPM for strongly nonlinear equation. In this study, it was observed that OHAM gives an accurate solution as compared to HPM. Moreover, an advantage of OHAM is that it does not need any initial guess or to identify the convergence control parameter-curve like Homotopy Analysis Method (HAM) and it is also parameter free. Furthermore, the OHAM has a built in convergence criteria similar to HAM but with a greater degree of flexibility [22]. Recently, from the best of our knowledge, three are no other researches attempt to solve nonlinear first order FIVPs OHAM. This method somewhat different from other approximate analytical methods in that it gives extremely good results for even a large domain with minimal terms of the approximate series solution. In OHAM, the control and adjust of the convergence region are provided in a convenient way. This method does not require linearization or discretization of the variables, and it is not affected by computation round off errors and one is not faced with necessity of large computer memory and time Moreover, OHAM is also parameter free and provides better accuracy over the approximate analytical methods at the same order of approximation. The outline of this paper is as follows: Section 2 describes the notations and basic definitions which are used in other sections. In Section 3 the defuzzification of first order nonlinear FIVP equation is presented with full details. In Section 4, we formulated and analyze the fuzzy OHAM to obtain a reliable approximate solution first order nonlinear FIVP. In Section 5, a numerical example for illustrating the capability and the flexibility of the proposed method is discussed. Finally, in Section 6 the conclusions of this study are given.

2. Preliminaries The definitions reviewed in this section are required in our work.

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Figure 1: Triangular Fuzzy Number

Definition 1. The link between the crisp and fuzzy domains h represented i ˜ by the r -level set (or r -cut set) of a fuzzy set A [21], denoted by A˜ which is r the crisp set of all x ∈ X such that µA˜ ≥ r, i.e. h i A˜ = x ∈ X µA˜ > r, r ∈ [0, 1] r

Definition 2. One of the important tools that uses to fuzzify the crisp models are fuzzy numbers which are subsets of the real numbers set, and represents vague values. Fuzzy numbers are linked to degrees of membership which state how true it is to say if something belongs or not to a determined set. A fuzzy number is called a triangular fuzzy number [22] defined by three numbers α < β < γ where the graph of µ(x) is a triangle with the base on the interval [α, β] and vertex at x = β and membership function has the following form:   0, if x < α.     x−α , if α ≤ x ≤ β, µ (x; α, β, γ) = β−α γ−x   γ−β , if β ≤ x ≤ y,   0, if x > γ.

(1)

where the r-level is defined by:

[µ (x)]r = [α + r (β − α) , γ − r (γ − β)] , r ∈ [0, 1] . ˜ and satisfy the The class of all fuzzy subsets of ℜ will be denoted by E following properties (see [23]): 1. µ (x) is normal, i.e ∃t0 ∈ ℜ with µ (t0 ) = 1,


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2. µ (x) Is convex fuzzy set, i. e. µ(λx + (1 − λ) y) ≥ min{µ (x) , µ(y)} ∀ x, y ∈ ℜ,λ[0, 1], 3. µ upper semi-continuous on ℜ, 4. {x ∈ ℜ:µ (x) > 0} is compact. ˜ is the space of fuzzy numbers and ℜ is a proper subset of E. ˜ where E Define the r -level set x ∈ ℜ, [µ]r = {x \ (x) ≥ r}, 0 ≤ r ≤ 1, where[µ]0 = {x \ (x) > 0} is compact which is a closed bounded interval and denoted by [µ]r = µ (x) , µ (x) . In the parametric form [24], which is represented by an ordered pair of functions µ (x; r) , µ (x; r) , r ∈ [0, 1] that satisfies the followings conditions: 1. µ (x; r) is a bounded left continuous non-decreasing function over [0, 1]. 2. µ (x; r) is a bounded left continuous non-increasing function over [0, 1] . 3. µ (x; r) ≤ µ (x; r) . A crisp number r is simply represented by µ (r) = µ (r) = r. ˜ for some interval T ⊆ E ˜ is called a Definition 3. A mapping f˜ : T → E fuzzy process or fuzzy function with crisp variable [25], and we denote r -level set by: [f˜(x)]r = f (x; r) , f (x; r) , x ∈ T, r ∈ [0, 1] ˜ be the set of all upper semi-continuous normal convexfuzzy numbers. where E Definition 4. Each function f : X → Y induces another function f˜ :

F (X) → F (Y ) defined for each fuzzy interval U in X by: ( Supx∈f −1 (y) U (x) , if y ∈ range (f ) , ˜ f (U ) (y) = 0, if y ∈ / range (f ) . This is called the Zadeh extension principle [26].

3. Defuzzification First Order Nonlinear FIVP Consider the general nonlinear first order FIVP involving ODE: e y˜′ (t) =Ae (t, y˜ (t)) + V(t),

y˜ (t0 ) =˜ y0 ,

t ∈ [t0 , T ] (2)

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where: y˜ (t) is a fuzzy function of the crisp variable t such that [˜ y (t)]r = [y(t; r), y(t; r)], Ae is fuzzy function of the crisp variable t and the fuzzy variable y˜, such that Ae (t, y˜ (t)) = a ˜f (t) + ˜bg (t) . (3) h i e y˜ (t)) = A(t, y˜ (t; r)), A(t, y˜ (t; r)) such that a ˜f (t) is the linear Here A(t; r term , ˜bg (t) is the nonlinear term:

where

h

˜ e (t) = c˜w (t) + d, V

(4)

i e V(t) = V(t; r), V(t; r) such that c˜w (t) + d˜ is the inhomogeneous r

term. The fuzzy coefficients in equations (3) and 4 are triangular fuzzy numbers denoted by h i ˜b = [b, b] , [˜ a]r = [a, a]r , r r

h i [˜ c]r = [c, c]r and d˜ = [d, d]r , r

for all r ∈ [0, 1]. y˜′ (t) is the first fuzzy H-derivative [22,24] such that

y˜′ (t) r = y ′ (t; r) , y ′ (t; r) ,

y˜0 is the are the fuzzy initialcondition and triangular fuzzy numbers denoted by [˜ y (t0 )]r = y (t0 ; r) , y (t0 ; r) . Also, we can write A(t, y˜ (t; r)) =F t, y, y r , (5) A(t, y˜ (t; r)) =G t, y, y r . Since y˜′ (t) = Ae (t, y˜ (t)) by using the extension fuzzy principle in Section 2, we can define the following membership function n o A(t, y˜ (t; r)) = min Ae (t, µ e(r) ∈ y˜(t; r) , e(r)) µ n o (6) A(t, y˜ (t; r)) = max Ae (t, µ e(r) ∈ y˜(t; r) , e(r)) µ


where

A(t, y˜ (t; r)) =F t, y (t; r) , y (t; r) = F (t, y˜ (t; r)) , A(t, y˜ (t; r)) =G t, y (t; r) , y (t; r) = G (t, y˜ (t; r)) .

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(7)

This defuzzification procedure is used in the fuzzification of the numerical and semi-analytical method for solving FDEs.

4. Fuzzification and Defuzzification of OHAM The formulations of OHAM for solving crisp differential equations was described in [20, 28]. In this section we present a new procedure for the approximate solution of linear and nonlinear first order FIVP by OHAM. Like in previous chapters we need to fuzzifying OHAM and then defuzzify it, by using the all fuzzy properties and definitions, including Zadeh extension principles in Section 3, we consider equation (2) followed by the defuzzification such that for all r ∈ [0, 1] we rewrite equation (2) in the following forms L y (t; r) − V(t; r) − F (t, y˜ (t; r)) =0, t ∈ [t0 , T ], (8) ∂[y]r B y (t; r) , =0, ∂t and for the upper bound we have L (y (t; r)) − V(t; r) − G (t, y˜ (t; r)) =0, t ∈ [t0 , T ], ∂[y]r B y (t; r) , =0. ∂t

(9)

According to OHAM that described in [28, 29], equations (8) and (9) can be written as follows: (1 − p) [L ([∅ (t; p)]r ) − V(t; r)] =H(p; r) [L ([∅ (t; p)]r ) − V(t; r) h i i −F e ∅ (t; p) , (10) h h i i h h r i (1 − p) L ∅ (t; p) − V(t; r) =H(p; r) L ∅ (t; p) − V(t; r) r r h i i −G e ∅ (t; p) , (11) r h i   h i ∂ e ∅ (t; p) r =0, (12) B e ∅ (t; p) , ∂t r

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∂ [∅(t;p)]r ∂[∅(t;p)]r ,L= are the linear operators and p ∈ [0, 1] is an where L = ∂t ∂t e embedding parameter. Here H (p; r) = H (p) , H (p) r is a nonzero auxiliary h i e r) = 0 and e fuzzy function for p6=0 and H(p; ∅ (t; p) is an unknown fuzzy

function. Obviously, when p = 0 and p = 1, we get: [∅ (t; 0)]r = y0 (t; r) , h i ∅ (t; 0) = y0 (t; r) , r

r

[∅ (t; 1)]r = y (t; r) , h i ∅ (t; 1) = y (t; r) .

(13)

r

h i Thus, as p increases from 0 to 1, the solution e ∅ (t; p) varies from y˜0 (t; r) to r

the solution of equations (10) –(12) where y˜0 (t; r) is obtained from equations (8) –(9) and for p = 0 we have: ∂[˜ y ] 0 r e r) = 0, B y˜0 (t; r) , = 0. (14) Le (˜ y0 (t; r)) − V(t; ∂t

e r) for equations (10) –(12) in the form: We choose auxiliary function H(p; 2

H (p; r) = C 1 (r)p + C 2 (r)p + · · · =

n X

C i (r)pi ,

i=1

2

H (p; r) = C 1 (r)p + C 2 (r)p + · · · =

n X

(15) C i (r)pi ,

i=1

where C˜1 (r) = C 1 (r) , C 1 (r) , C˜2 (r) = C 2 (r) , C 2 (r) , . . . are the constants that become a function of r for all r ∈ [0, 1]. Expanding hto be found i e ˜ ∅ t; pCi (r) into Taylor’s series about p, we obtain approximate solution r series: n h i X e ˜ (16) yi t, Cˇ (r) pi . e ∅ t; p, Ci (r) = y˜0 (t; r) +

r

i

r

i=1

Now, substitute equation (16) into (10) and (11) then collecting the coefficient of like powers of p, we obtained the following linear equations. The zeroth order problem is given by (14), and the first and second order problems are given as follows: First order problem: L y 1 (t; r) − V(t; r) =C 1 (r) F0 ( y˜0 (t; r)) , (17) L ( y 1 (t; r)) − V(t; r) =C 1 (r) G0 ( y˜0 (t; r)) ,

57


∂[˜ y1 ]r B y˜1 (t; r) , ∂t

= 0.

(18)

Second order problem L y 2 (t; r) − L y 1 (t; r) = C 2 (r) F0 ( y˜0 (t; r)) h i + C 1 (r) L y 1 (t; r) + F1 ( y˜0 (t; r) , y˜1 (t; r))

(19)

L ( y 2 (t; r)) − L ( y 1 (t; r)) = C 2 (r) G0 ( y˜0 (t; r)) + C 1 (r) L ( y1 (t; r)) + G1 ( y˜0 (t; r) , y˜1 (t; r)) , ∂[˜ y2 ] r =0 B y˜2 (t; r) , ∂t

(20)

The general n-th order formula with respect to yñ (t; r) is given by: L

y n (t; r) −L y n−1 (t; r) = C n (r) F0 ( y˜0 (t; r))    n−1 n−1 X X C i (r) L y n−i (t; r) + Fn−i  + y j (t; r) i=1

j=0

L ( y n (t; r)) − L yn−1 (t; r) = C n (r) G0 ( y˜0 (t; r))    n−1 n−1 X X C i (r) L y n−i (t; r) + Gn−i  y j (t; r), +

(21)

j=0

i=1

∂[˜ yn ]r B yñ (t; r) , ∂t

=0

(22)

P n−1 y ˜ (t; r) are the coefficient pn of y ˜ (t; r) and G n−i j j j=0 j=0 h i h i in the expansion of F e ∅ (t; p) and G e ∅ (t; p) about the embedding parameter r r p " ! !# ! n n ∞ X X X e ∅ t; p, F [˜ y i ] pn , Fn C i (r) =F0 (˜ y (t; r)) + where Fn−i

P n−1

0

G

" e ∅ t; p,

i=1 n X i=1

r

!# !

C i (r)

r

=G0 (˜ y 0 (t; r)) +

r

n=1 ∞ X

n=1

Gn

i=0 n X i=0

[˜ yi ] r

!

(23)

pn .

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It has been observed that the convergence of the series (16) depends upon the auxiliary constants C˜1 (r), C˜2 (r), . . . , then at p = 1, we obtain: y˜∗ t,

n X

C˜i (r); r

i=1

!

= y˜0 (t; r) +

n X i=1

"

y˜i (t,

#

n X

C˜i (r)))

i=1

(24) r

Substituting (24) into (8) and (9) it results the following residual: R t,

n X

C i (r) ; r

i=1

!

=L y∗

R t,

C i (r) ; r

i=1

!

C i (r); r

i=1

+F n X

t,

n X

y˜∗ t,

n X

C˜i (r); r

i=1

=L y∗

t,

n X

C i (r); r

i=1

+G

y˜∗

t,

n X

!!

i=1

!!

!!

C˜i (r); r

+ V(t; r) , (25)

+ V(t; r)

!!

.

e = 0, then y˜∗ yields the exact solution, in general it doesn’t happen, espeIf R cially in nonlinear FIVPs. For the determinations auxiliary constants of C˜i (r), i = 1, 2. . . n, we choose t0 and T regarding to equation (2) such that optimum values of f C i (r) for the convergent solution of the desired problem is obtained. To find the optimal values of C˜i (r) here, we apply the method of Least Squares as follows: ! ! Z n n b X X C˜i (r); r dt (26) R2 t, C˜i (r); r = Se t, i=1

a

i=1

e is the residual where R

and

h i − V(t; r) − F ([˜ y∗ ]r ) , [R]r =L y ∗ r R r =L ([y ∗ ]r ) − V(t; r) − G ([˜ y∗ ]r ) , ∂ Se ∂ Se ∂ Se = = ... = 0. ∂ C˜1 (r) ∂ C˜2 (r) ∂ Cñ (r)

(27)

(28)

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5. Results and Discussions In this section we employ OHAM first order nonlinear FIVP to show the efficiency of this method. We compare the result of these problems solved by OHAM with HPM [9]. Also, we analyze and discuss the results obtained by OHAM in details Consider the first order fuzzy Riccati equation [9] y˜′ (t) = [˜ y (t)]2 + t2 ,

t ≥ 0,

y˜ (0) = [0.1r − 0.1, 0.1 − 0.1r] ,

(29)

∀ r ∈ [0, 1].

In this example we use another way of OHAM to obtain the approximateanalytical solution. We can construct fifth order OHAM series solution with ˜ only one convergence control parameter C1 (r) . This way can obtained the same results with five convergence control parameters C˜ 1 (r) , . . . , C˜5 (r) . Also, this way provide less time of CPU to run especially in FIVPs which for some r -level sets some value of the convergence control parameters are missing or equation (29) need long time to run[20, 29]. Now we can contract fifth order of OHAM series solution for all r ∈ [0, 1] of equation (29) as follows: Zero order problem: y˜0′ (t; r) − t2 = 0,

y˜0 (0; r) = [0.1r − 0.1, 0.1 − 0.1r] .

First order problem y˜1′ t, C˜1 (r); r = 1 + C˜1 (r) ey ′0 (t; r) − 1 + C˜1 (r) t2 − C˜1 (r)( ey0 (t; r))2 ,

(30)

(31)

y˜1 (0; r) =0.

Second order problem y˜2′ t, C˜1 (r); r = 1 + C˜1 (r) ey′1 (t; r)

− 2C˜1 (r) ey 0 (t; r) ey 1 (t; r) ,

(32)

y˜2 (0; r) =0.

Third order problem y˜3′ t, C˜1 (r); r = 1 + C˜1 (r) ey ′2 (t; r) − 2C˜1 (r) ey 0 (t; r) ey 2 (t; r) − C˜1 (r)( ey 1 (t; r))2 ,

y˜3 (0; r) =0.

(33)

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Fourth order problem y˜4′ t, C˜1 (r); r = 1 + C˜1 (r) ey ′3 (t; r)

− 2C˜1 (r) ( ey 1 (t; r) ey 2 (t; r) − ey 0 (t; r) ey 3 (t; r)) ,

(34)

y˜4 (0; r) =0.

Fifth order problem y˜5′ t, C˜1 (r) ; r = 1 + C˜1 (r) ey′3 (t; r)

− 2C˜1 (r) ( ey 1 (t; r) ey 3 (t; r) + ey 0 (t; r) ey 4 (t; r)) − C˜1 (r) ( ey 2 (t; r))2 ,

(35)

y˜5 (0; r) =0.

Then the fifth order OHAM approximate solution is given by y˜∗ (t; r) =˜ y0 (t; r) + y˜1 t, C˜ 1 (r); r + y˜2 t, C˜1 (r; r + y˜3 t, C˜1 (r) ; r + y˜4 t, C˜1 (r; r + y˜5 t, C˜1 (r) ; r .

(36)

Since equation (29) is without exact solution and in order to detect the accuracy of OHAM solution we need to define the residual error [30] as follows h i 2 2 ˜ OHAM = y˜′ (t; r) − [˜ E y (t; r)] − t (37) ∗ ∗ r

The solutions of the lower and upper bounds of equations (30–35) are obtained by using Mathematica 10 Package. From Table 1 one can see the fifth order OHAM series solution at t = 0.5 and for all 0 ≤ r ≤1 that satisfy the fuzzy numbers properties in Section 2 by taking the triangular shape. From table (2) we conclude that the accuracy of the approximate solution of equation (29) obtained using 5-terms of OHAM is better than of the fifth order HPM approximate solution in [9] for all r ∈ [0, 1] when t = 0.5.

6. Conclusion In this article, we studied formulate and applied the optimal homotopy asymptotic method in finding an approximate solution first order nonlinear fuzzy

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OHAM ® t=0

yHt;rL

OHAM ® t=0.1

yHt;rL

0.10

0.10

0.05

0.05

0.2

0.4

0.6

0.8

1.0

r-level

0.2

-0.05

-0.05

-0.10

-0.10

0.4

OHAM ® t=0.2

yHt;rL

0.6

0.8

1.0

r-level

OHAM ® t=0.3

yHt;rL

0.10 0.10

0.05

0.05

0.2

0.4

0.6

0.8

1.0

r-level 0.2

-0.05

0.4

0.6

0.8

1.0

0.8

1.0

r-level

-0.05

OHAM ® t=0.4

yHt;rL

OHAM ® t=0.5

yHt;rL 0.15

0.10 0.10 0.05 0.05

0.2

0.4

0.6

0.8

1.0

r-level 0.2

0.4

0.6

r-level

-0.05 -0.05

Figure 2: Approximate solution of fifth order OHAM in the form of triangular fuzzy number for all r ∈ [0, 1]

4. ´ 10 -10

0.00

1.0

OHAM Solution

Residual Error

1.0

3. ´ 10 -10 2. ´ 10 -10

-0.05 1. ´ 10 -10 0

-0.10 0.0

0.5

0.2

r-level

0.5

0.0

r-level

0.2

t

t 0.4

0.4

0.0

0.0

Figure 3: Approximate solution and accuracy of fifth order OHAM solution of equation (29) for all in three dimensional form for all r ∈ [0, 1] and t ∈ [0, 0.5]

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r

C 1 (r)

y (0.5; r) OHAM

C 1 (r)

y (0.5; r) OHAM

0 0.2 0.4 0.6 0.8 1

−9.82761 × 10−1 −9.93047 × 10−1 −9.99561 × 10−1 −1.02109 −1.02962 −1.03595

−5.44361 × 10−2 −3.59324 × 10−2 −1.70686 × 10−2 2.16584 × 10−3 2.17819 × 10−2 4.17911 × 10−2

−1.04533 −1.04824 −1.04395 −1.04904 −1.05168 −1.03595

1.4817 × 10−1 1.26006 × 10−1 1.043 × 10−1 8.30373 × 10−2 6.22055 × 10−2 4.17911 × 10−2

Table 1: The optimal values of the convergence control parameter of fifth order OHAM solution for equation (29) at t = 0.5 for all 0 ≤ r ≤ 1.

r

E (0.5; r) OHAM

E (0.5; r) HP M [6]

E (0.5; r) OHAM

0 0.2 0.4 0.6 0.8 1

1.346854 × 10−9 7.045693 × 10−10 1.549903 × 10−10 8.581980 × 10−11 1.171446 × 10−11 3.831896 × 10−11

1.630679 × 10−6 1.034307 × 10−6 6.334864 × 10−7 3.513160 × 10−7 1.349086 × 10−7 5.463861 × 10−8

2.673856 × 10−11 7.610135 × 10−11 3.355999 × 10−11 3.784562 × 10−11 1.276814 × 10−10 3.831896 × 10−11

E (0.5; r) HP M [9] 1.994205 × 10−6 1.279712 × 10−6 8.091260 × 10−7 4.871649 × 10−7 2.504207 × 10−7 5.463861 × 10−8

Table 2: Accuracy of fifth order OHAM solution of equation (29) at t = 0.5 for all 0 ≤ r ≤ 1

initial value problems. To the best of our knowledge, this is the first attempt for solving first order nonlinear fuzzy initial value problems with OHAM. In OHAM, the control and adjustment of the convergence of the series solution using the convergence control parameters are achieved in a simple way in a numerical example including first order fuzzy Riccati equation showing the capability and efficiency of the OHAM. We obtained accurate solution without the need to compare with the exact solution by using even low order approximation. The numerical results obtained by OHAM satisfy the fuzzy number properties by taking the convex fuzzy number shape. Moreover the procedure of OHAM has advantages over some existing analytical approximation methods.

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