Solving Fuzzy Duffing’s Equation by the Laplace

Applied Mathematical Sciences, Vol. 6, 2012, no.59, 2935 - 2944 Solving Fuzzy Duffing’s Equation by the Laplace Transform Decomposition ‫܌܉ܕܐۯ ܑܖ܉’ܚܗܗۼ‬૚ , Mustafa ‫ ܜ܉ܕ܉ۻ‬૛ , J. ۹‫ ܚ܉ܕܝܓܑܞ܉‬૜ and Nor ‫ ܚܑܕۯ ܐ܉܌ܑܛܕ܉ܐ܁‬۶‫ܐ܉ܢܕ܉‬૝ 1,3,4 Department of Science and Mathematics, Faculty of Science Arts and Heritage, Universiti Tun Hussein Onn Malaysia 86400 Parit Raja, Johor, Malaysia. Department of Mathematics, Faculty of Science and Technology Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Malaysia [email protected] Abstract The Laplace transform decomposition algorithm (LTDA) is a numerical algorithm which can be adapted to solve the Fuzzy Duffing equations (FDE). This paper will describe the principle of LTDA and discusses its advantages. Concrete examples are also studied to show the numerical results on how LTDA efficiently work to solve the Duffing equations in the fuzzy setting. Keywords: Laplace transform decomposition algorithm, differential equations, Fuzzy Duffing equations, numerical algorithm, Maple. 2 1 Introduction Various problems involving nonlinear control systems in physics, engineering and communication theory, makes increasing interest to study the periodically forced Duffing's equation [6, 10-12]. In many cases, the Duffing’s equations have their uncertainty known as fuzzy Duffing’s equation (FDE). The Adomian decomposition method (ADM) can be used to find the solutions of a large class of nonlinear problems [1, 5, 7, 8, 13, 14]. However, the implementation of the ADM mainly depends upon the calculation of Adomian polynomials for the nonlinear operators and their numerical solutions were obtained in the form of finite series. The linear term y(x) represents an infinite sum of components ‫ݕ‬௡ ሺ‫ݔ‬ሻ.2. ‫ݕ‬ଶ . 7]. the initial values in the form of fuzzy numbers will be transformed in the parametric forms. we bring some basic definitions of fuzzy numbers and how the LTDA is applied in solving FDE. number in [0. … . in order to solve the FDE. we have applied the Laplace. By using the Maple software. The Adomian decomposition method has been proved to be reliable. The decomposition method consists of splitting the given equation into linear and nonlinear terms. differential.2936 Noor'ani Ahmad et al The present paper aims at offering an alternative method to get solution of the FDE by using the Laplace transform numerical scheme on the basis of the decomposition method [12] named Laplace transform decomposition method (LTDM). we discuss the conclusion and future research. ‫ݕ‬௡ . accurate and effective in both the analytic and the numerical purposes [2. 1] for the membership function of ‫ ܣ‬evaluates at x. ‫ݕ‬ሺ‫ ݔ‬ሻ ൌ ෍ ‫ݕ‬௡ ௡ୀ଴ ௄ This paper is organized as follows. etc. a sign over a letter to denote a fuzzy subset of the real numbers. In this case. integral. Fuzzy numbers are considered because many situations in real-world are facing uncertainty and vagueness [3. In this study. 2 Preliminaries This section will begin with defining the notation used in this paper. In section 3. the truncated sum is calculated.1. … defined by ‫ݕ‬ሺ‫ ݔ‬ሻ ൌ ෍ ‫ݕ‬௡ ሺ‫ݔ‬ሻ ௡ୀ଴ ∞ The decomposition method identifies the nonlinear term ‫ܨ‬ሺ‫ݕ‬ሺ‫ ݔ‬ሻሻ by the decomposition series ‫ܨ‬ሺ‫ݕ‬ሺ‫ ݔ‬ሻሻ ൌ ෍ ‫ܣ‬௡ ௡ୀ଴ ∞ where ‫ܣ‬௡ s are Adomian’s polynomial of ‫ݕ‬଴ . with fuzzy initial value problems. ‫ݕ‬ଵ . In section 4. partial differential. In section 2. An iterative algorithm is achieved for the determination of ‫ݕ‬௡ ‫ ݏ‬in recursive manner. We write ‫ܣ‬ ሚ ሚ. the algorithm is practically applied in relation to the numerical examples. and the numerical results show that the LTDA approximates the exact solution with a high degree of accuracy using only few terms of the iterative scheme. The decomposition method has been introduced by Adomian [9] to solve linear and nonlinear functional equations such as in algebraic. 4]. We place a ~ ሚ ሺ‫ݔ‬ሻ. ݊ ൌ 0. An α-cut of ‫ܣ‬ . 5) using the initial condition (2. For arbitrary fuzzy numbers x =(‫( ݔ‬α). x + y = ( ‫( ݔ‬α ) + ‫( ݕ‬α ). we obtain ‫ ݏ‬ଶ ࣦሾ‫ݕ‬ሿ െ ‫ݕ‬ሺ0ሻ‫ ݏ‬െ ‫ ݕ‬′ ሺ0ሻ ൅ ‫݌‬ሼ‫ࣦݏ‬ሾ‫ݕ‬ሿ െ ‫ݕ‬ሺ0ሻሽ ൅ ‫݌‬ଵ ࣦሾ‫ݕ‬ሿ ൅ ‫݌‬ଶ ࣦሾ‫ ݕ‬ଷ ሿ ൌ ࣦሾ݂ሺ‫ݔ‬ሻሿ. 3. ‫( ̅ݔ‬α ) + ‫ ݕ‬ ഥ (α )).1) ‫ݕ‬ሺ0ሻ ൌ ൫∝ . ܽଷ ሻ and ߚ ൌ ሺܾଵ . ݇ ൒ 0. ‫݌‬ଵ . y = ( ‫( ݕ‬α ). ‫( ݔ‬α) ) . The Laplace transform Decomposition Method is applied to find the solution to the following nonlinear fuzzy initial value problems: ‫ ݕ‬′′ ൅ ‫ ݕ݌‬′ ൅ ‫݌‬ଵ ‫ ݕ‬൅ ‫݌‬ଶ ‫ ݕ‬ଷ ൌ ݂ሺ‫ݔ‬ሻ. we obtain ࣦሾ‫ ݕ‬′′ ሿ ൅ ‫ࣦ݌‬ሾ‫ ݕ‬′ ሿ ൅ ‫݌‬ଵ ࣦሾ‫ݕ‬ሿ ൅ ‫݌‬ଶ ࣦሾ‫ ݕ‬ଷ ሿ ൌ ࣦሾ݂ ሺ‫ ݔ‬ሻሿ (2. (2. 0 ≤ α ≤ 1. ‫( ݕ‬α )) and real number k. 1. ݇‫ ݔ‬ൌ ቊ ൫݇‫ݔ‬.3) By using linearity of Laplace transform. for 0 ൏ ߙ ൑ 1 and α–cuts of fuzzy written by ‫ܣ‬ numbers are always closed and bounded. ݇‫ ݔ‬൯.2) where ‫݌‬. ∝൯‫ ݌‬െ ‫݌‬ଵ ࣦሾ‫ݕ‬ሿ െ ‫݌‬ଶ ࣦሾ‫ ݕ‬ଷ ሿ ൅ ሺࣦሾ݂ሺ‫ݔ‬ሻሿ ሺ2. ܾଶ . ߚቁ ൅ ൫∝. 0 ≤ α ≤ 1. ‫( ݑ‬α ) ≤ ‫( ݑ‬α ). ൫݇‫ ݔ‬.2). If ∝ൌ ሺܽଵ . ‫ݑ‬ሺߙሻቁ. ߚ ቁ ൌ ሺܾଵ ൅ ሺܾଶ െ ܾଵ ሻ‫ ݎ‬. We represent an arbitrary fuzzy number Applying the formulas on Laplace transform. then the parametric forms for and ߚ are ൫∝ .Solving fuzzy Duffing’s equation 2937 by an ordered pair of functions ቀ‫ݑ‬ሺߙሻ. ܽଷ ൅ ሺܽଶ െ ܽଷ ሻ‫ݎ‬ሻ and ቀߚ . ݇‫ ݔ‬൯. ∝ ൯ ൌ ሺܽଵ ൅ ሺܽଶ െ ܽଵ ሻ‫ ݎ‬. The method consists of applying Laplace transformation (denoted throughout this paper by ࣦሾ‫ ݕ‬′′ ሿ ൅ ࣦሾ‫ ݕ݌‬′ ሿ ൅ ࣦሾ‫݌‬ଵ ‫ݕ‬ሿ ൅ ࣦሾ‫݌‬ଶ ‫ ݕ‬ଷ ሿ ൌ ࣦሾ݂ሺ‫ݔ‬ሻሿ. we have ሺ‫ ݏ‬ଶ ൅ ‫ݏ݌‬ሻࣦሾ‫ݕ‬ሿ ൌ ൫∝. Let F be the set of all upper semi-continuous normal convex fuzzy numbers with bounded α-level sets. (2. 2. A crisp number α is simply represented by ‫( ݑ‬α ) = ‫( (ݑ‬α ) = α . ܾଷ ൅ ሺܾଶ െ ܾଷ ሻ‫ݎ‬ሻ respectively. 2. ܽଶ . ݇ ൏ 0. (2.6ሻ or .1]. ∝ ൯ and ‫ ݕ‬′ ሺ0ሻ ൌ ቀߚ . ∝൯‫ ݏ‬൅ ቀߚ. and ‫݌‬ଶ are real constants and ∝ and ߚ in parametric forms. ߚ ቁ. ‫( ݑ‬α ) is a bounded left continuous non increasing function over [0.4) 3. ‫( ݑ‬α ) is a bounded left continuous non decreasing function over [0. ሚ ሺߙሻ. which satisfies the following requirements: 1. x = y if and only if ‫( ݔ‬α )= ‫( ݕ‬α ) and ‫( ݔ‬α ) = ‫( ݕ‬α ).1]. (2. ܾଷ ሻ.is defined as ൛‫ݔ‬ห‫ܣ‬ ሚ ൒ ߙൟ. (2. ∝ ‫ ݏ‬൅ ߚ ൅∝ ‫݌‬ ‫݌‬ଵ ‫݌‬ଶ ࣦ ቂ‫ݕ‬ቃ ൌ െ ଶ ࣦሾ‫ݕ‬ሿ െ ଶ ࣦሾ‫ ݕ‬ଷ ሿ ଶ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ 1 ൅ ଶ ࣦሾ݂ሺ‫ݔ‬ሻሿ. ሺ2. ‫ܣ‬ଵ ൌ ‫ݕ‬ଵ ݄ଵ ሺ‫ݕ‬଴ ሻ. ሺ2. ∝ሻ‫݌‬ ‫ݏ‬ଶ ൅ ‫ݏ݌‬ 1 ൅ ଶ ࣦሾ݂ሺ‫ݔ‬ሻሿ.7ሻ ‫ ݏ‬൅ ‫ݏ݌‬ െ ‫ݏ‬ଶ ‫݌‬ଵ ‫݌‬ଶ ࣦሾ‫ݕ‬ሿ െ ଶ ࣦሾ‫ ݕ‬ଷ ሿ ൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ In other words Eq. ‫ݕ‬௡ and are calculated by the formula 1 ݀௡ ‫ܣ‬௡ ൌ ሾ ௡ ሾ݄ ෍ ߣ௜ ‫ݕ‬௜ ሿሿ ݊! ݀ߣ ∞ ௜ୀ଴ ݄ሺ‫ݕ‬ሻ ൌ ‫ ݕ‬ൌ ෍ ‫ܣ‬௡ . ሺ2. ሺ2.11ሻ ఒୀ଴ . ଶ ଺ and for ݄ሺ‫ݕ‬ሻ ൌ ‫ ݕ‬ଷ they are given by ‫ܣ‬଴ ൌ ‫ݕ‬଴ ଷ . Also the nonlinear operator ݄ሺ‫ݕ‬ሻ ൌ ‫ ݕ‬ଷ is decomposed as ଷ ∞ ௡ୀ଴ ௡ୀ଴ ∞ where the ‫ܣ‬௡ s are Adomian polynomials of ‫ݕ‬଴ .2938 Noor'ani Ahmad et al ࣦሾ‫ݕ‬ሿ ൌ ൫∝. in particular ‫ ݕ‬ൌ ෍ ‫ݕ‬௡ . ሺ2.9ሻ ‫ ݏ‬൅ ‫ݏ݌‬ The Laplace transform decomposition technique consists next of representing the solution as an infinite series. 1. … . 2. … ሺ2. ݊ ൌ 0.13) . 2 ଵ ଵ ‫ܣ‬ଷ ൌ ‫ݕ‬ଷ ݄ଵ ሺ‫ݕ‬଴ ሻ ൅ ‫ݕ‬ଵ ‫ݕ‬ଶ ݄ଶ ሺ‫ݕ‬଴ ሻ ൅ ‫ݕ‬ଵ ଷ ݄ଷ ሺ‫ݕ‬଴ ሻ. 1 ‫ܣ‬ଶ ൌ ‫ݕ‬ଶ ݄ଵ ሺ‫ݕ‬଴ ሻ ൅ ‫ݕ‬ଵ ଶ ݄ଶ ሺ‫ݕ‬଴ ሻ.7) are in the fuzzy forms written as lower and upper cases. ‫ܣ‬ଵ ൌ 3‫ݕ‬଴ ‫ݕ‬ଵ . ߚቁ ൅ ሺ∝. (2. ‫ݕ‬ଵ .10ሻ where the term ‫ݕ‬௡ are to recursively calculated.8ሻ ‫ ݏ‬൅ ‫ݏ݌‬ ∝ ‫ ݏ‬൅ ߚ ൅∝ ‫݌‬ ‫݌‬ଵ ‫݌‬ଶ ࣦሾ‫ݕ‬ሿ ൌ െ ଶ ࣦሾ‫ݕ‬ሿ െ ଶ ࣦሾ‫ ݕ‬ଷ ሿ ଶ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ 1 ൅ ଶ ࣦሾ݂ሺ‫ݔ‬ሻሿ.12ሻ The first few polynomials are given by ‫ܣ‬଴ ൌ ݄ሺ‫ݕ‬଴ ሻ. ∝ ൯‫ ݏ‬൅ ቀߚ. the results are: ∞ ∞ ∞ ∝ ‫ ݏ‬൅ ߚ ൅∝ ‫݌‬ ‫݌‬ଵ ‫݌‬ଶ ࣦ ൥෍ ‫ݕ‬൩ ൌ െ ଶ ࣦ ൥෍ ‫ݕ‬൩ െ ଶ ࣦ ൥෍ ‫ܣ‬௡ ൩ ‫ ݏ‬ଶ ൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ௡ୀ଴ ௡ୀ଴ ௡ୀ଴ 1 ൅ ଶ ࣦሾ݂ሺ‫ݔ‬ሻሿ ሺ2.16ሻ ‫ ݏ‬൅ ‫ݏ݌‬ Matching both sides of (2. and (2. ‫ݏ‬ଶ For upper case: ∝ ‫ ݏ‬൅ ߚ ൅∝ ‫݌‬ 1 ࣦൣ‫ݕ‬଴ ൧ ൌ ൅ ଶ ࣦሾ݂ሺ‫ݔ‬ሻሿ ሺ2. ሺ2. ‫݌‬ଵ ‫݌‬ଶ ࣦ ቂ‫ݕ‬௡ାଵ ቃ ൌ െ ଶ ࣦ ቂ‫ݕ‬௡ ቃ െ ଶ ࣦൣ‫ܣ‬௡ ൧.17ሻ ଶ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫݌‬ଵ ‫݌‬ଶ ࣦ ቂ‫ݕ‬଴ ቃ – ଶ ࣦൣ‫ܣ‬଴ ൧.14) Substituting (2.15ሻ ‫ ݏ‬൅ ‫ݏ݌‬ ∞ ∞ ∞ ∝ ‫ ݏ‬൅ ߚ ൅∝ ‫݌‬ ‫݌‬ଵ ‫݌‬ଶ ࣦ ൥෍ ‫ݕ‬൩ ൌ െ ଶ ࣦ ൥෍ ‫ݕ‬൩ െ ଶ ࣦ ൥෍ ‫ܣ‬௡ ൩ ଶ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ௡ୀ଴ ௡ୀ଴ ௡ୀ଴ 1 ൅ ଶ ࣦሾ݂ሺ‫ݔ‬ሻሿ. .10). ሺ2. the following iterative algorithms are obtained.15) and (2.20ሻ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫݌‬ଵ ‫݌‬ଶ ࣦൣ‫ݕ‬଴ ൧ െ ଶ ࣦൣ‫ܣ‬଴ ൧.22ሻ ൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫݌‬ଵ ‫݌‬ଶ ࣦൣ‫ݕ‬ଶ ൧ ൌ െ ଶ ࣦൣ‫ݕ‬ଵ ൧ െ ଶ ࣦൣ‫ܣ‬ଵ ൧.11) to (2. ‫ܣ‬ଷ ൌ 3‫ݕ‬଴ ‫ݕ‬ଷ ൅ 6‫ݕ‬଴ ‫ݕ‬ଵ ‫ݕ‬ଶ ൅ ‫ݕ‬ଵ ଷ . ሺ2.19ሻ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ . ሺ2. 2939 (2.Solving fuzzy Duffing’s equation ‫ܣ‬ଶ ൌ 3‫ݕ‬଴ ଶ ‫ݕ‬ଶ ൅ 3‫ݕ‬଴ ‫ݕ‬ଵ ଶ . . ࣦൣ‫ݕ‬ଵ ൧ ൌ െ .23ሻ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ .16).8) and (2.9) respectively.18ሻ ࣦ ቂ‫ݕ‬ଵ ቃ ൌ െ ଶ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫݌‬ଵ ‫݌‬ଶ ࣦ ቂ‫ݕ‬ଶ ቃ ൌ െ ଶ ࣦ ቂ‫ݕ‬ଵ ቃ െ ଶ ࣦൣ‫ܣ‬ଵ ൧. . ሺ2. ሺ2.21ሻ ଶ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ . For lower case: ∝ ‫ ݏ‬൅ ߚ ൅∝ ‫݌‬ 1 ࣦ ቂ‫ݕ‬଴ ቃ ൌ ൅ ଶ ࣦሾ݂ሺ‫ ݔ‬ሻሿ ሺ2. … ௜ୀ଴ ௜ୀ଴ ௞ 3 Numerical example The Laplace transform decomposition algorithm is illustrated by the following example [8]. we can express ݂ሺ‫ݔ‬ሻ in Taylor series at ‫ݔ‬଴ ൌ 0.2940 ࣦൣ‫ݕ‬௡ାଵ ൧ ൌ െ Noor'ani Ahmad et al ‫݌‬ଵ ‫݌‬ଶ ࣦൣ‫ݕ‬௡ ൧ െ ଶ ࣦൣ‫ܣ‬௡ ൧. ‫ݕ‬ଶ ሻ. Example. ሺ2.21) we obtain the value of ሺ‫ݕ‬଴ . ‫ݕ‬ଷ .1ሻ with initial conditions ‫ݕ‬ ෤ሺ0ሻ ൌ ሺെ1. ܽ௜ ൌ ݂ ௜ୀ଴ ௞ ݂ ሺ௜ሻ ሺ0ሻ .2ሻ ሺ3.2.1. … . However.27ሻ ଶ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ݏ‬ ‫ ݏ‬൅ ‫ݏ݌‬ By using LTDA which is described above.20) and (2.3ሻ By using ADM. ‫ݕ‬ଶ ሿ.24ሻ ൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ݏ‬ଶ Applying the inverse Laplace transform to (2. and then applying the inverse Laplace transform. we evaluating the Laplace transform of the quantities on the right side of ࣦሾ‫ݕ‬ଵ .18) and (2. Substituting these values of ‫ݕ‬଴ to (2. ݅ ൌ 0. in this study our initial value problem will be written in the parametric forms of the fuzzy setting and the equation is then solved by LTDM.1ሻ ‫ݕ‬ ෤ ′ ሺ0ሻ ൌ ሺ0. Since the complicated excitation term ݂ሺ‫ݔ‬ሻ can cause difficult integrations and proliferation of terms.17) and (2.25ሻ ݅! Eq. If we replace ݂ሺ‫ݔ‬ሻ by ሚሺ‫ݔ‬ሻ ൌ ෍ ܽ௜ ‫ ݔ‬௜ . ‫ݕ‬଴ ሻ. Consider the Duffing’s equation in the following type: ‫ ݕ‬′′ ൅ 3‫ ݕ‬െ 2‫ ݕ‬ଷ ൌ cos ‫ ݔ‬sin 2‫ ݔ‬ሺ3.3) was found.26ሻ ଶ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ ݏ‬൅ ‫ݏ݌‬ ‫ݏ‬ ∝ ‫ ݏ‬൅ ߚ ൅∝ ‫݌‬ 1 ܽ௜ ݅! ࣦൣ‫ݕ‬଴ ൧ ൌ ൅ ଶ ෍ ௜ାଵ ሺ2.0. this example was considered in [11] and its solution (3. (2.1. … in lower and upper cases can be obtained recursively in a similar way by using (2. we obtain the values of (‫ݕ‬ଵ .21) become ௞ ∝ ‫ ݏ‬൅ ߚ ൅∝ ‫݌‬ 1 ܽ௜ ݅! ࣦ ቂ‫ݕ‬଴ ቃ ൌ ൅ ଶ ෍ ௜ାଵ ሺ2.2ሻ The analytic solution of this equation is ‫ݕ‬ሺ‫ݔ‬ሻ ൌ sinሺ‫ݔ‬ሻ ሺ3. ሺ2.17) and (2. ‫ݕ‬ ෤ଶ .24) respectively. which is truncated for simplification. . ݇.22) respectively. The other terms ‫ݕ‬ଶ . ‫ݕ‬ ෤ଵ . we obtain values ‫ݕ‬ ෤଴. we obtain 1 7 61 ଻ 547 ‫ݕ‬଴ ൌ െ0.584 2.7) we get ‫ݕ‬ଵ ൌ 0.14ሻ Operating with Laplace inverse on both sides of (3.11ሻ ‫ݏ‬ଷ ‫ݏ‬ ‫ݏ‬ ‫ݏ‬ ‫ݏ‬ Substituting (3.0016‫ ଼ ݔ‬െ 0.0058‫ ݔ‬ଵ଴ ൅ 0. ሺ3.0184‫ ଼ ݔ‬൅ ⋯ ሺ3.064‫ ݔ‬ସ ൅ 0.6ሻ ࣦሾ‫ݕ‬ ෤଴ ሿ ൌ ଶ ൅ ଶ ࣦൣ݂ ‫ݏ‬ ‫ݏ‬ 3 2 ሚ ଴ ൧.13) yields ‫ݕ‬ଶ ൌ 0. ‫݌‬ଶ ൌ െ2.624 15.612 2.8ሻ ࣦሾ‫ݕ‬ ෤ଶ ሿ ൌ െ ଶ ࣦሾ‫ݕ‬ ෤ଵ ሿ ൅ ଶ ࣦൣ‫ܣ‬ ‫ݏ‬ ‫ݏ‬ in general.7).0228‫ ݔ‬ହ ൅ 0. then the result is 0.10ሻ 3 60 2520 181440 ଷ Substituting this value of ‫ݕ‬଴ and ‫ܣ‬଴ ൌ ‫ݕ‬଴ given in (2.03662‫ ଻ ݔ‬൅ 0. 1ሻ.0668‫ ଺ ݔ‬െ 0.8 for the lower case.4ሻ 3 60 2520 Eq.0175‫ ଻ ݔ‬െ 0.12) to (3.14) to (3. (3. ‫݌‬ଵ ൌ 3.8‫ ݔ‬൅ ‫ ݔ‬ଷ െ ‫ ݔ‬ହ ൅ ‫ ݔ‬െ ‫ ݔ‬ଽ ሺ3.292‫ ݔ‬ଶ െ 0.0052‫ ݔ‬ହ ൅ ⋯ ሺ3.5). ሺ3.108 184.2).0. considering ‫ ݌‬ൌ 0. For example ‫ݕ‬ଷ ൌ 0. ∝ൌ ሺെ1. ሺ3.2 ൅ 0.8) and using ‫ܣ‬ଵ given in (2.9ሻ ࣦሾ‫ݕ‬ ෤௡ାଵ ሿ ൌ െ ଶ ࣦሾ‫ݕ‬ ෤௡ ሿ ൅ ଶ ࣦൣ‫ܣ‬ ‫ݏ‬ ‫ݏ‬ Substituting the inverse Laplace transform to (3. we obtain following iterative algorithm: 1 1 ሚ൧.36 ࣦ ቂ‫ݕ‬ଵ ቃ ൌ െ ସ െ ହ ൅ ଺ െ ଻ ൅ ⋯ ሺ3.1) becomes ሚሺ‫ ݔ‬ሻ ሺ3.7ሻ ࣦሾ‫ݕ‬ ෤ଵ ሿ ൌ െ ଶ ࣦሾ‫ݕ‬ ෤଴ ሿ ൅ ଶ ࣦൣ‫ܣ‬ ‫ݏ‬ ‫ݏ‬ 3 2 ሚଵ ൧.0070‫ ݔ‬ଽ ൅ 0. 3 2 ሚ ௡ ൧.2ሻ to Eq.14).208 1. ሺ3.6) for the r =0.15ሻ ‫ݕ‬ସ ൌ 0.1. ሺ3.0097‫ ݔ‬ଽ ൅ ⋯ ሺ3. (3.16ሻ .0028‫ ଺ ݔ‬൅ 0.12ሻ Higher iterations can be easily obtained by using the Maple software.5ሻ ‫ ݕ‬′′ ൅ 3‫ ݕ‬െ 2‫ ݕ‬ଷ ൌ ݂ with fuzzy initial condition (3.13ሻ ‫ݏ‬ ‫ݏ‬ ‫଻ݏ‬ ‫଼ݏ‬ ‫ݏ‬ଽ The inverse Laplace transform applied to (3. we obtain 1.583 742.994 ࣦሾ‫ݕ‬ଶ ሿ ൌ െ ହ ൅ ଺ ൅ െ ൅ ൅ ⋯ ሺ3.Solving fuzzy Duffing’s equation 2941 If we replace 7 61 ହ 547 ଻ ሚሺ‫ ݔ‬ሻ ൌ 2‫ ݔ‬െ ‫ ݔ‬ଷ ൅ ݂ ‫ ݔ‬െ ‫ ݔ‬.368‫ ݔ‬ଷ െ 0.0292‫ ଼ ݔ‬൅ 0.0022‫ ݔ‬ଵଵ ൅ ⋯ ሺ3.536 0. ߚ ൌ ሺ0.730 48. If we reapply the given above algorithm. 118‫ ଻ ݔ‬൅ ⋯ ሺ3.0011‫ ݔ‬ଽ െ 0.24ሻ 2 20 840 60480 3 ହ 3 523 ଽ 18281 ଵଵ ‫ݕ‬ ෤ଶ ൌ ‫ ݔ‬െ ‫ ଻ݔ‬െ ‫ ݔ‬൅ ‫ ݔ‬െ ⋯ ሺ3.27ሻ 4480 246400 5491200 8072064000 Also calculated the sum of five terms for 1 1 ହ 1 1 ߶ହ ሺ‫ ݎ‬ൌ 1.570796327 in Eq.17).1312‫ ݔ‬ସ ൅ ⋯ ሺ3. we have the sum of five terms ଶ for r = 0.552‫ ݔ‬ଷ ൅ 0.0347‫ ݔ‬ଷ െ 0. particularly the sum of five terms is calculated as below: ߶ହ ሺ‫ ݎ‬ൌ 0. we always deal with ambiguities condition for example .22). we have the sum of five terms ଶ for r = 1.28ሻ 6 120 5040 362880 గ Substituting ‫ ݔ‬ൌ ൌ 1.25ሻ 40 40 20160 2217600 3 ଻ 29 ଽ 859 92901 ‫ ݔ‬൅ ‫ ݔ‬൅ ‫ݔ‬ଵଵ െ ‫ ݔ‬ଵଷ ൅ ⋯ ሺ3.0239‫ ݔ‬ଽ ൅ ⋯ ሺ3.8ሻ ൌ 0.0262‫ ଻ ݔ‬൅ 0.0013‫ ݔ‬ଵ଴ ൅ 0.2 ൅ 1.2 ൅ 1.8ሻ ൌ െ0.20ሻ ‫ݕ‬ସ ൌ 0.2‫ ݔ‬൅ ‫ ݔ‬ଷ െ ‫ ݔ‬൅ ‫ ݔ‬െ ‫ ݔ‬ଽ ሺ3.8 is 1.2920‫ ݔ‬ଶ െ 0.067‫ ݔ‬ସ ൅ 0.150‫ ଺ ݔ‬െ 0.26ሻ ‫ݕ‬ ෤ଷ ൌ െ 560 960 739200 12812800 1 1843 ଵଵ 20287 ଵଷ 23386949 ଵହ ‫ݕ‬ ෤ସ ൌ ‫ݔ‬ଽ െ ‫ ݔ‬൅ ‫ ݔ‬൅ ‫ ݔ‬൅ ⋯ ሺ3.144‫ ݔ‬ସ ൅ 0.0038‫ݔ‬ଵଵ ൅ ⋯ ሺ3.80253.20ሻ ‫ݕ‬ଷ ൌ െ0.22ሻ గ By using Maple. … as below: 1 7 ହ 61 ଻ 547 ‫ݕ‬଴ ൌ 0.2112‫ ݔ‬ସ ൅ ⋯ ሺ3.034‫ ݔ‬ହ െ 0. In the real-world problems. 4 Conclusion In this paper.2‫ ݔ‬െ 0. (3.570796327 in Eq. we have series of ‫ݕ‬଴ .2187‫ ݔ‬ଷ ൅ 0.23ሻ 3 60 2520 181440 1 1 47 ଻ 89 ‫ݕ‬ ෤ଵ ൌ െ ‫ ݔ‬ଷ ൅ ‫ ݔ‬ହ ൅ ‫ ݔ‬െ ‫ ݔ‬ଽ െ ⋯ ሺ3. (3. investigation on solving FDE by LTDM has been done and exists. By using the same method in iterations for the upper case.84425.0 is 1.0000.0657‫ ଼ ݔ‬൅ 0.2942 Noor'ani Ahmad et al Substituting ‫ ݔ‬ൌ ൌ 1.0001‫ ଼ ݔ‬െ 0.2920‫ ݔ‬ଶ െ 0.18ሻ 3 60 2520 181440 ‫ݕ‬ଵ ൌ െ0. ‫ݕ‬ଵ . Meanwhile for r = 1.8‫ ݔ‬൅ 0.21ሻ Also calculated the sum of five terms: ߶ହ ሺ‫ ݎ‬ൌ 0. we have the sum of five terms ଶ for r = 0. ‫ݕ‬ଶ .8 is 0.28).0ሻ ൌ െ ‫ ݔ‬ଷ ൅ ‫ ݔ‬െ ‫ ଻ݔ‬൅ ‫ ݔ‬ଽ ൅ ⋯ ሺ3.19ሻ ‫ݕ‬ଶ ൌ 0.2 ൅ 0. In this case. it is easier to get the determinate partial sum ߶௡ ሺ‫ ݔ‬ሻ ൌ ∑௡ ௠ୀ଴ ‫ݕ‬௠ .292‫ ݔ‬ଶ െ 0.17ሻ గ Substituting ‫ ݔ‬ൌ ൌ 1.570796327 in Eq. the results obtained are as below: 1 7 61 ଻ 547 ‫ݕ‬ ෤଴ ൌ ‫ ݔ‬൅ ‫ ݔ‬ଷ െ ‫ ݔ‬ହ ൅ ‫ ݔ‬െ ‫ ݔ‬ଽ ሺ3.0028‫ ଺ ݔ‬൅ 0. (3.127‫ ݔ‬ହ ൅ ⋯ ሺ3. 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