Variational principle for some nonlinear problems

A variational principle is established by the semi-inverse method and used to solve approximately a nonlinear problem by the Ritz method. In this process,it may be difficult to solve a large system of algebraic equations,the Groebner bases theory (Buchberger’s algorithm) is applied to solve this problem. The results show that the variational approach is much simpler and more efficient.


Introduction
With the rapid development of nonlinear science,various kinds of analytical methods were used to handle nonlinear problems, such as the homotopy perturbation method (He , 2002(He , 2004(He , 2009(He , 2014Anjum and He 2020a, b;Yu et al. 2019;Ren et al. 2019;El-Dib 2020a, b, 2021), variational iteration method (He , 2007(He , 2011(He , 2012aHe and Wu 2007;Liu et al. 2021), Taylor series method (He and Ji 2019;He 2019aHe , 2020a, Exp-function method (He and Wu 2006;He 2013;, and variationalbased methods (He 2020bHe and Ain 2020;Wu 2021;Wang and Wei 2021). Recently, there have been some reasearch results in the field of differential equations, for example, in Wang et al. (2020), the ZIR of the fractal RC circuit are modeled by LFD,where the transient local fractional ordinary differential equation is obtained with aid of the law of switch and Kirchhoff Voltage Laws. In Cao and Dai (2021), Dai B Yi Tian ttxsun@163.com 1 and Wang (2020), Liu et al. (2020), Yu et al. (2020), Wu et al. (2020), new methods are presented to find traveling wave solutions of PDEs. In addition,the author has done some work on nonlinear problems, e.g. exact solutions (Tian 2019a(Tian , b, 2016 and numerical solutions (Tian 2018;Tian and Yan 2016) of integral and differential equations.
Each method has its advantages and disadvantages. For example, the Taylor series method is simple, but it is low convergence hinders its wide applications.The Expfunction method can lead to the analytical solutions, but its complex calculation makes those inaccessible who are not familiar some mathematics software. The variationalbased methods can obtain a globally valid solution, however, it is extremely difficult to establish a needed variational principle for a complex nonlinear problem, etc..
The variational principle play a important role in nonlinear science,especially for differential equations. For example, it can provides the conservation laws in an energy form, and reveals the possible solution structures of PDEs (He 2019b). In this paper, the semi-inverse method (He 2003a, b) is used to search variational principle of differential equations, then the Ritz method is effectively and conveniently used. In this process, it may be difficult to solve a large system of algebraic equations,the Groebner bases theory (Buchberger's algorithm) is applied to solve this problem.
The rest of this paper is organized as follows. In Sect. 2, we briefly review the basic Groebner Bases theory. In Sect. 3, Variational approach to some boundary value problems are studied. In Sect. 4, He's semi-inverse method is applied to solve some nonlinear problems. In Sect. 5, we conclude this paper and some discussions are also given there.

Groebner Bases
In the following, we list the basic results of Groebner Bases theory (Cox et al. 2007).
Definition 2 Let f 1 , . . . , f s be polynomials in k[x 1 , . . . , x n ]. Then we set The crucial fact is that f 1 , . . . , f s is an ideal.

Definition 3
We will write f F for the remainder on division of f by the ordered s- . . , f s ). If F is a Groebner basis for ( f 1 , . . . , f s ),then we can regard F as a set (without any particular order) by Proposition I . Definition 4 Let I ⊂ k[x 1 , . . . , x n ] be an ideal. We will denote by V(I ) the set V(I ) = {(a 1 , . . . , a n ) ∈ k n : f (a 1 , . . . , a n ) = 0 for all f ∈ I }.

ODEs boundary value problems
Example 1 Consider the following ODE (Lao 2015): with the boundary conditions where y is the differentiation with respect to x. By the semi-inverse method, the variational principle of Eq.
(1) reads applying Ritz's method, we choose a trial function satisfying all the boundary conditions.
the approximate solution can be expressed as Case 1. n = 1.
where a is a unknown constant to be further determined.
Substituting (6) into (3) yields making J n=1 stationary with respect to a results in we have a = 5 9 , and the first-order approximate solution of Eq. (1) is Case 2. n = 2.
where a, b are unknown constants to be determined. Substituting (10) into (3) yields making J n=2 stationary with respect to a, b results in we have a = 142 369 , b = 14 41 , and the second-order approximate solution of Eq. (1) is Case 3. n = 3.
where a, b, c are unknown constants to be determined. Substituting (15) into (3) yields making J n=3 stationary with respect to a, b, c results in Equation (17) sometimes is large and difficult to be solved by hand, even by software,such as MAPLE and MATLAB. In this paper,we use Buchberger's algorithm to solve this problem. Let I be the ideal corresponding to the original system of Eq. (17), and we want to find all points in V(I ).
Using Buchberger's algorithm with the lex order a > b > c,we find a Groebner basis: thus,there are three solutions altogether of g 1 = g 2 = g 3 = 0, which are since V(I ) = V(g 1 , g 2 , g 3 ),we have found all solutions of the original equations (17), and the third-order approximate solution of Eq. (1) is Example 2 Consider the following example (He 2003b): subject to the boundary conditions By the semi-inverse method,the variational principle of Eq. (24) reads applying Ritz's method, we choose a trial function the approximate solution can be expressed as Case 1. n = 1.
where a is a unknown constant to be further determined. Substituting (30) into (27) yields making J n=1 stationary with respect to a results in we have a = 180(e − 3), and the first-order approximate solution of Eq. (24) is Case 2. n = 2.
where a, b are unknown constants to be determined. Substituting (34) into (27) yields making J n=2 stationary with respect to a, b results in we have and the second-order approximate solution of Eq. (24) is Case 3. n = 3.
where a, b, c are unknown constants to be determined. Substituting (40) into (27) yields Let I be the ideal corresponding to the system of Eq. (42). Using Buchberger's algorithm with the lex order a > b > c,we find a Groebner basis: thus, and the third-order approximate solution of Eq. (24) is Remark 1 The solutions of Eq. (24) we obtained are not presented in (He 2003b).

PDEs boundary value problems
Example 3 Consider the following elliptic equation: subject to the boundary conditions By the semi-inverse method,the variational principle of Eq. (49) reads applying Ritz's method, we choose a trial function satisfying all the boundary conditions.
the approximate solution can be expressed as Case 1. n = 1.
where a is a unknown constant to be further determined.
Substituting (54) into (51) yields making J n=1 stationary with respect to a results in we have a = 5 7 , and the first-order approximate solution of Eq. (49) is Case 2. n = 2.
where a, b, c are unknown constants to be determined. Substituting (63) into (51) yields Let I be the ideal corresponding to the system of Eq. (65). Using Buchberger's algorithm with the lex order a > b > c, we find a Groebner basis: thus, and the third-order approximate solution of Eq. (49) is

He's semi-inverse method
For a given nonlinear partial differential equation the main steps of this method are as follows (Elboree 2015; Najafi and Arbabi 2016): Step 1. seek solitary solutions of Eq. (72) by taking u(x, t) = u(ξ ), ξ = x − ct,and transform Eq. (72) to the ordinary differential equation where prime denotes the derivative with respect to ξ .
Step 2. If possible,integrate Eq. (73) term by term one or more times, for simplicity, the integration constants set to zero.
Step 3. According to He's semi-inverse method, we construct the following trialfunctional where L is a Lagrangian for Eq. (73).
Step 4. By the Ritz method, we can obtain different forms of solitary wave solutions, such as u(ξ ) = A · sech(Bξ), u(ξ ) = A · csch(Bξ), u(ξ ) = A · tanh(Bξ), u(ξ ) = A · coth(Bξ) and so on.In this paper, we search a solitary wave solution in the form where A and B are constants to be further determined. Substituting Eq. (75) into Eq. (74) and making J stationary with respect to A and B results in solving simultaneously Eqs. (76) and (77), we obtain A and B. Hence, the solitary wave solution Eq. (75) is well determined.

The PHi-4 equation
The PHi-4 equation play an important role in nuclear and particle physics over the decades. Let us consider the Phi-4 equation in the form (Akter and Akbar 2015): where m and λ are real valued constants. Using the traveling wave variable ξ = x − ct, Eq. (78) is transformed into the following ODE: by He's semi-inverse method,we can obtain the following variational formulation: Case A: we search for a soliton solution in the form by substituting (81) into (80), we obtain to find the constant A,we need to solve the following equation: form Eq. (83),we obtain therefore,the solitary wave solutions to the PHi-4 equations are constructed as follows: Case B: we search for a soliton solution in the form by substituting (86) into (80), we obtain to find the constant A,we need to solve the following equation: form Eq. (88),we obtain therefore,the solitary wave solutions to the PHi-4 equations are: By a similar manipulation,we have: and so on. Case C: we search for a soliton solution in the form by substituting (95) into (80), we obtain to find the constant A,we need to solve the following equation: form Eq. (97),we obtain to find the constant A,we need to solve the following equation: form Eq. (107),we obtain therefore,the solitary wave solutions is constructed as follows: Case B: suppose soliton solution in the form by substituting (110) into (103), we obtain to find the constant A,we need to solve the following equation: form Eq. (112), we obtain therefore,the solitary wave solutions is: u(ξ ) = 1 4 5k 2 − 9 · sech 2 (ξ ).

Conclusion
In this paper,based on the semi-inverse method, variational formulation for some nonlinear problems are established, and the approximate solutions are easily obtained by the Ritz method. In this process, it may be difficult to solve a large system of algebraic equations, the Groebner bases theory (Buchberger's algorithm) is applied to solve this problem. At present, the application of Groebner bases theory maily on