Homotopy analysis method for fuzzy Boussinesq equation

In this work, the fuzzy Boussinesq equation is considered to solve via the homotopy analysis method (HAM). For this purpose, a theorem is proved to illustrate the convergence of the proposed method. Also, two sample examples are solved by applying the HAM to verify the efficiency and importance of the method.


Introduction
In recent years, some numerical and analytical methods were proposed in order to solve fuzzy differential equations [1-8, 10, 18, 19]. One of the powerful semi-analytical methods to solve differential equations is the homotopy analysis method (HAM). In [14], the authors applied this method to solve the Boussinesq equation in crisp case. In this work, we consider the fuzzy form of Boussinesq equation as follows: e u tt þ ae u xx þ bðe u 2 Þ xx À e u xxxx ¼ e 0; 0 t T; x [ 0: where e u is unknown fuzzy function, a and b are crisp constant coefficients and e f and e g are known fuzzy functions.
In order to solve Eq. (1), we apply the HAM in fuzzy case as an important and efficient method to find the solution of differential equations. The HAM, proposed by Liao, [16,17], is a semi-analytical method which the solution is obtained as a series form according to a recursive relation stems from a deformation equation [13,14].
In Sect. 2, we remind some fuzzy concepts briefly. In Sect. 3, we apply the HAM to solve the fuzzy Boussinesq equation and we prove a theorem to show the convergence of the proposed method. In Sect. 4, we solve two sample fuzzy Boussinesq equations and we obtain a series solution by this method.

Remark 2.9
Note that by the above definition, a fuzzy function is i-differentiable or ii-differentiable of order n if f ðsÞ for s ¼ 1; . . .; n is i-differentiable or ii-differentiable. It is possible that the different orders have different kind i or ii differentiability.

Main idea
In order to describe the HAM for Eq. (1), we consider the following equation: According to the parametric form of fuzzy numbers, we consider Eq. (4) in the following form: where I is the identity matrix, L ¼ is an auxiliary function matrix, is an auxiliary parameter matrix, /ðx; t; QÞ ¼ /ðx; t; r; qÞ /ðx; t; r; qÞ is an unknown function matrix, u 0 ðx; t; rÞ is an initial guess of the vector matrix which denotes the embedding parameter matrix. It is obvious, when the q, increases from 0 to 1 or in other word the embedding parameter matrix changes from Q ¼ 0 to Q ¼ I, the solution of system of equations (5) changes from /ðx; t; r; 0Þ ¼ u 0 ðx; t; rÞ to /ðx; t; r; IÞ ¼ uðx; t; rÞ: Therefore, /ðx; t; rÞ varies from the initial guess u 0 ðx; t; rÞ to the exact solution u(x, t, r) of the system. We consider /ðx; t; r; QÞ in the following matrix expansion form, The convergence of the vector series (6) The m-th order deformation system can be written as where If we consider L ¼ Theorem 2.10 If the series solution (8) of problem (1) obtained from the HAM and also the series Proof Without loss of generality, we suppose u be i-differentiable with respect to the x, t and also it be a positive fuzzy number (8t 2 ½0; T). Therefore, we can write Eq. (1) in the following form: e u tt þ ða þ À a À Þe u xx þ ðb þ À b À Þðe u 2 Þ xx À e u xxxx ¼ e 0; where a þ ; a À ; b þ ; b À ! 0. Therefore, we have N½uðx; t; rÞ N½uðx; t; rÞ We write X n m¼1 ½u m ðx; t; rÞ À v m u mÀ1 ðx; t; rÞ ½R m ð u ! mÀ1 Þ: Since h 6 ¼ 0 and Hðx; tÞ 6 ¼ 0, we have From (12), it holds We consider P þ1 Similarly for next elements. Finally, and it means that e u tt þ ae u xx þ bðe u 2 Þ xx À e u xxxx ¼ e 0: h

Test examples
In this section, we solve two sample examples to illustrate the applicability of the proposed method. The results are provided by Maple. where e 0 ¼ ð3r À 3; 3 À 3rÞb u xx þ ðr 2 À ð3 À 2rÞ; ð3 À 2rÞ 2 À ðrÞÞb u xxxx and b u is the solution of crisp case of the equation. We suppose, e u t be i-differentiable with respect to the t and e u x ; e u 2 x and e u xxx are i-differentiable with respect to the x. Also e u be a positive fuzzy number (8t 2 ½0; T; x [ 0), therefore we have u tt À u xx þ ðu 2 Þ xx À u xxxx À ð3r À 3Þb u xx À ðr 2 À ð3 À 2rÞÞb u xxxx u tt À u xx þ ðu 2 Þ xx À u xxxx À ð3 À 3rÞb u xx À ðð3 À 2rÞ 2 À rÞb u xxxx We consider H ¼ I and h ¼ ÀI; and also, we choose the initial approximate as Therefore, In general, the series solution is given by That gives the exact solution Therefore, e u ¼ ðr; 3 À 2rÞ 6 ðxþtÞ 2 is the exact solution of the fuzzy differential equation.
We suppose e u t be ii-differentiable with respect to the t, e u x ; e u xxx are ii-differentiable with respect to the x and e u 2 x is i-differentiable with respect to the x. Also e u be a negative fuzzy number (8t 2 ½0; T; x [ 0), therefore we have We consider H ¼ I and h ¼ ÀI; and also, we choose the initial approximate as Therefore, In general, the series solution is given by That gives the exact solution Therefore, e u ¼ r 2 þ 1 2 ; 2 À r À Á À6 ðxþtÞ 2 is the exact solution of the fuzzy differential equation.

Conclusion
In this work, we applied the fuzzy HAM in order to solve the fuzzy Boussinesq equation. For this aim, we considered the parametric form of a fuzzy number and established the deformation equations for two crisp Bousinnesq equations obtained from the proposed method. Also, we presented a theorem to warrant the convergence of the proposed method too. Similar to the discussion in this work, the HAM can be used in order to solve other kinds of fuzzy differential equations as an efficient and proper method.