Solving Variable-Coefficient Fourth-Order ODEs with Polynomial Nonlinearity by Symmetric Homotopy Method

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1 Applied and Computational Mathematics 218; 7(2): doi: /j.acm ISSN: (Print); ISSN: (Online) Solving Variable-Coefficient Fourth-Order ODEs with Polnomial Nonlinearit b Smmetric Homotop Method Abdrhaman Mahmoud 1, 2, Bo Yu 1, Xuping Zhang 1, * 1 School of Mathematical Sciences, Dalian Universit of Technolog, Dalian, China 2 Department of Mathematics, Facult of Sciences and Technolog, Omdurman Islamic Universit, Omdurman, Sudan address: * Corresponding author To cite this article: Abdrhaman Mahmoud, Bo Yu, Xuping Zhang. Solving Variable-Coefficient Fourth-Order ODEs with Polnomial Nonlinearit b Smmetric Homotop Method. Applied and Computational Mathematics. Vol. 7, No. 2, 218, pp doi: /j.acm Received: Februar 4, 218; Accepted: Februar 24, 218; Published: March 22, 218 Abstract: In this paper, the eigenfunction epansion method (EEM) is applied to find numerical solutions for variablecoefficient fourth-order ordinar differential equations (ODEs) with polnomial nonlinearit. The smmetr of the solution set for the resulting sstem of polnomial equations obtained from EEM of the problem is analzed. The smmetric homotop method is constructed to calculate all solutions of the discretization sstem for the problem. Due to the eploitation of smmetr, the number of computations is reduced. Numerical eamples are presented to demonstrate the efficienc of the presented homotop method. Kewords: Fourth-Order ODEs, Sstem of Polnomial Equations, Homotop Continuation Method, Numerical Algebraic Geometr, Smmetr Group 1. Introduction It is known that the nonlinear differential equations can govern man phenomena in nature. Once the nonlinearit increases in these equations, the structures of the solutions ma become complicated. Due to these nonlinearities, the ODEs cannot be solved b using analtical methods. Therefore, the numerical methods can be used to solve such equations. Thus, the approimate solutions are required. Boundar value problems (BVPs) for nonlinear higherorder ODEs have significant applications in applied mathematics. Especiall, nonlinear fourth-order BVPs are commonl used in a wide variet of application fields such as phsics, chemical phenomena, and engineering; see for eample [1, 2]. Recentl, a great attention has been given to solve these problems numericall. In this paper, we consider the following tpe of variablecoefficient fourth-order ODEs with polnomial nonlinearit ;, Ω,1, (1) with boundar conditions 1 1, where and are given continuous functions on the interval,1. ; is a polnomial of with given variable coefficients,,,, and degree. The BVP (1), usuall describes the deformation of an elastic beam, which has been widel studied b man researchers [3-6]. In general, the eistence and multiplicit of solutions to such kind of BVPs depend on the growth conditions of the nonlinearit term ; [7-1]. The EEM is one of the numerical techniques for finding multiple solutions of differential equations when some analtical methods fail. Hence, in this paper, we use the EEM as a discretization method, which is more accurate than other numerical methods such as finite difference and finite element methods [11]. Under the assumptions that and are in!,1, the linear fourth-order operator in BVP (1) has an infinite set of real eigenvalues " # $ # $ # & $, and the corresponding eigenfunctions '( )! * satisfies,

2 Applied and Computational Mathematics 218; 7(2): ( #(, + ( = (1 = ( = ( 1 = It can be verified that the eigenfunctions are ( =,../., where, is an arbitrar constant. After normalization, these eigenfunctions are a normalized orthogonal base of the Sobolev space 1:= 3 Ω 3 Ω. The normalized eigenpairs of BVP (1) are (2) ( = 2./.,# =. 7,. = 1,2,3. Let 1 & denote the 9-dimensional subspace spanned b the first 9 eigenfunctions. The approimate solution of (1) can be obtained b the linear combination of '( ) & * as follows: find & = & ; ( 1 &, such that * < &,( = = & ( + & ( + & ( The approimation epressed b (3) can be written as; & = ; & ( =, ( 1 & (3)? 2@ A; B C. D./../D+.D;E.;ED+./../D B* = 2= ; &./., 1.,D 9 (4) From the above epression, we get the following sstem of polnomial equations with respect to ; = ;,;,,; & F & ;,c,,; & ; I = ( where = &? B* ; B ( = &? B* ; B ( ( = &? B* ; B ( ( =, (5) I = =. D./../D+.D;E.;ED+./../D. The integral of. D./../D in (4) can be computed eplicitl while other integral terms ma not be computed eplicitl. We can compute them b using the numerical quadrature. Given that 9 is large, the sstem of polnomial equations (5) will be a complicated sstem. Therefore, we can solve this sstem b using the numerical approimation techniques such as Fied-point iteration method, Newton s method and Homotop continuation method [12]. The main purpose of this paper is to construct a smmetric homotop for solving variable-coefficient fourth-order ODEs with polnomial nonlinearit. We construct a simple fourthorder ODE as a starting sstem and then discretize it in eigensubspaces, where the subsstems have readil available solutions. Then, the resulting sstems of polnomial equations in eigensubspaces are put together in a block-wise manner to construct the starting sstem for a general problem. We use the smmetr to reduce the number of computations b tracking the representative solution paths of the homotop. For spurious solutions which ma appear in the solution set of the discretized sstem of BVPs, we removed them b using certain filters. The remainder of this paper is organized as follows. In section 2, we introduce a homotop continuation method for the resulting sstem of polnomial equations obtained from the discretization of BVPs. In section 3, we address the analsis of smmetr group in the solution set of discretized problem. Section 4 contains the construction of smmetric homotop for solving variable-coefficient fourth-order ODEs with polnomial nonlinearit. In section 5, we provide numerical eamples to verif the efficienc of the smmetric homotop. Finall, conclusions are summarized in section Homotop Continuation Method for Sstem of Polnomial Equations It is known that Newton s method ma not converge when solving a large sstem of polnomial equations that arise in connection with the discretization of BVPs. Sometimes, the computation of associated Jacobian matri is rather epensive. Therefore, it is necessar to provide a substitute such as a homotop continuation method, which has global convergence properties [13-15]. The basic idea of using a homotop continuation method is to deform a simple starting sstem to a target sstem and track the zero-dimensional (all isolated) solutions of the intermediate sstems. For a given sstem of polnomial equations, we construct an appropriate start sstem which can be easil solved and then deform these solutions through the smooth paths (homotop paths) to get the desired solutions of the target sstem. From the numerical algebraic geometr communit, there are some software packages such as PHCpack, HOM4PS-2., and Bertini [16-18]. These software packages can be used to solve the given sstem of polnomial equations. Therefore, the sstem of polnomial equations (5) can be solved b using the following total degree homotop 3 & ;,c,,; &,J = 1 JK & ;,c,,; & +JF & ;,c,,; & (6)

3 6 Abdrhaman Mahmoud et al.: Solving Variable-Coefficient Fourth-Order ODEs with Polnomial Nonlinearit b Smmetric Homotop Method where K & ;,;,,; & is the starting sstem defined as follows; ; N 1 K & ;,c,,; & LM Q, = degf, ; P O 1 moreover L C a generic random number is used to avoid the singularities [19]. Recentl, a few studies are dealing with computing the solutions of the discretization differential equations b using the homotop continuation method; see, for eample [11, 2-22] and the references therein. Allgower et al. suggested a numerical continuation method in [2] to compute all solutions for a nonlinear second order two-point BVPs b using a finite difference method. The performed a homotop deformation on successivel refined discretization sstems to obtain solutions on the finer level. When the sstem of polnomial equations is large, the proposed some filters for removing spurious solutions, which leads to an efficient homotop. These filters depend on the information and properties of the solutions of the original problem, and there seems no general rule. For the smmetr of solution set for the discretized sstem, a special filter is selected. Zhang et al. [11], proposed the EEM to obtain multiple solutions of semilinear elliptic partial differential equations with polnomial nonlinearit. The computed the corresponding solutions for the discretized problem on a coarse level, then utilize it as initial guesses to calculate related solutions of the discretized problem on a finer level. The etension homotop method was constructed to find all solutions of the resulting sstem of polnomial equations efficientl. The used the error estimates of EEM to propose a filter strateg for removing spurious solutions. The finite element Newton method was applied to refine the computed solutions more. In [21], Zhang et al. designed the smmetric homotop method to find solutions of the sstem of polnomial equations derived from the discretizations of elliptic equations with cubic and quintic nonlinearities. The analzed the smmetr of solution set for the sstem of polnomial equations arising from the eigenfunction epansion discretization of the problem. This smmetr arises from the dihedral smmetr V 7 of the unit square. The proved that this kind of homotopies could preserve the smmetr and reduce the number of computations, because onl the representative paths have to be traced. The homotop method introduced in [22] is a bootstrapping approach, which was applied to compute multiple solutions of differential equations. This method used a homotop continuation method based on the domain decomposition. That means to decompose the domain into subdomains, and then each subdomain is solved independentl in parallel. Then, the solutions from the subdomains are combined to build solutions for the original problem. The applied this approach for solving problems consisting one and two-dimensional problems. 3. Smmetr Group for the Solution Set of the Discretized Problem Throughout this section, we discuss the smmetr of discretized problem (1) b using the group actions of the dihedral group V (the point group). The dihedral group V consists of the identit and a reflection about the center of the domain; L =, L = 1, L V (7) The smmetr of the discretized solution set is due to the V smmetr of the domain, and it is passed over through the eigenfunctions, the eigen-base and the epansion coefficients. We will address them in more details as follows: First, recall that the eigenfunctions of the linear fourth-order operator with boundar conditions have the following form ( = 2./.,. = 1,2,3 For L V, the transformation on eigenfunctions can be obtained b; For eample, if L = 1, we get L ( = ( L. (8) L ( = = = 1 X 2./. = 1 X ( Therefore, the transformation on eigen-base can be defined in 1 & as follows; g Z?(,,( =?L (,,L ( (9) For eample, again if L = 1, we get g Z?(,(,( [,( = (, (,( [, ( 7 Note that, the representation of g Z is an orthogonal matri. That means for an element in R & as a column vector, the transformation g Z corresponds to ] Z can be written as follows; I I 1 I M I I Q ] Z M [ I Q = M 1 I QM [ 1 I Q [ I 7 I 7 1 I 7 Second, the transformation on epansion coefficients. For & = & ; ( 1 &, define :1 & R & as; * I & = ;,;,,; & _. (1)

4 Applied and Computational Mathematics 218; 7(2): The transformation g`z on epansion coefficients induced b is defined as & L & This leads to the following result ; ; L a(,(,,( & b Since g Z is an orthogonal matri, we have As a result, we find that ; & de = (,(,,( & ag`z b g`z = L f. ; ; ; & de (11) g Z?(,(,,( =?(,(,,( Z _ =?(,(,,( Z f (12) ; ; L a(,(,,( & b Again, this leads to the following result; ; & de = gg Z (,(,,( & hab g`z = g Z f ; ; ; & de = (,(,,( & a] Z f b ; ; ; & de (13) Finall, the transformation on coefficients of the discretized problem with coefficients of nonlinearit ; can be as; = & ( & ( & ( = & ( & ( & ( = = L & = = L +? & + & (14) + +? & (15) The smmetr in the solutions of sstem of polnomial equations F & ;,;,,; & defined in (5) can be stated as follows; Let = L and define Then, Ck & (k +k & (k +k & (k Besides that, k = L L =, L V (16) = C & ( + & ( + & ( = Cg?L ( +L &.L ( +L &.L ( h = ;k & (k = = ; & ( Therefore, the resulting polnomial sstem F & ;,;,,; & becomes = =?;L L ( (17) F & ;,;,,; & = g Z F & mg`z ;,;,,; & n = g`zf F & g`z ;,;,,; & (18) Let o & denotes the solution set of F & =. The epansion coefficients ;,;,,; & is said to be equivalent to ;,;,,; & if, for some g Z V, ;,;,,; & = g Z ;,;,,; & Therefore, we can divide the solution set b assigning o & to the equivalence classes. These equivalence classes under the equivalence relation are known as orbits. The set of representative solutions is a set of all equivalence classes in o &, i.e., o & /. So, computing all solutions of F & = depends on the computing all representative solutions, then we perform group actions on them.

5 62 Abdrhaman Mahmoud et al.: Solving Variable-Coefficient Fourth-Order ODEs with Polnomial Nonlinearit b Smmetric Homotop Method 4. Construction of the Smmetric Homotop for Discretized Problem In this section, we focus on the construction of the smmetric homotop for variable-coefficient fourth-order ODEs with polnomial nonlinearit. Due to the smmetr analzed in Section 3, we used it to construct an efficienc homotop for our problem; and we limit our stud to the nonlinearit case = 3,5. Recall that the eigenvalues of the linear fourth-order ODE operator are s s s &, and the corresponding eigenfunctions are t,t,,t &. Denote us = v/t, where t = 2./.. Let w z :1 us denotes the -orthogonal projection which is defined as;? w z,t@ =, t us. For L V, consider the transformation on -orthogonal projection is L?w z = w z L & 4.1. The Smmetric Homotop for Cubic Polnomial Nonlinearit Consider the following simple case of fourth-order BVP as the starting problem for our general problem with polnomial nonlinearit = 3 + = [,,1 = 1 = = 1 = (19) The EEM for (19) in an eigensubspace corresponding to an eigenvalue can be written as follows: find = ; t us, such that = & t = = [ & t, t us (2) 2. 7 ; C./. = ; [ 2 7 C./ 7.,. = 1,2,,9 (21) Note that the variables ; in the discretized problem (21) are separable and each equation has 3 solutions. Therefore (21) has 3 & real solutions.,, is the trivial solution. For nontrivial solutions, suppose that the nonzero components are located at.,,. {, we have eplicit epression for these components as follows; ; N = ; = = ; } = z }.~ (22) Thus, we find that it is eas to solve the discretized problem and then can set the resulting sstem of polnomial equations (21) block-wise to obtain the starting sstem for the general problem. B using the smmetr, we can classif the solutions of the resulting sstem of polnomial equations obtained from EEM into the equivalence classes. Therefore, we onl need to focus on the calculation of the representative solutions. Thus, we construct the following smmetric homotop for our problem with cubic nonlinearit where K & is defined as follows; 3 & ;,;,,; &,J = 1 JK & ;,;,,; & +JF & ;,;,,; &, (23) K & ;,;,,; & C?w z t and, C is the generic random number for avoiding the singularities The Additional Smmetr for Odd Cubic Nonlinearit It is known that the additional smmetr in the solutions of (1) can be obtained when ; is an odd function of. That means if & is a solution of (1), then & is also a solution, which is named Z smmetr. Therefore, we denote b V Z the smmetr group of the discretized solutions and the homotop solution paths The Smmetric Homotop for Quintic Polnomial Nonlinearit Analogues to cubic polnomial nonlinearit, the following simple case of fourth-order BVP can be the starting problem for our general problem with polnomial nonlinearit = 5 + = ~,,1 = 1 = = 1 = (24) C?w z [ t =,. = 1,2,,9 Again, the EEM for (24) in an eigensubspace corresponding to an eigenvalue can be written as follows: find = ; t us, such that = & t 5) = = ~ & t, t us ( ; C./. = ; ~ 2 C./.,. = 1,2,,9 (26) Similarl, the variables ; in the discretized problem (26) are separable and each equation has 5 solutions. Therefore (26) has 5 & solutions.,, is the trivial solution. For nontrivial solutions, the nonzero components are

6 Applied and Computational Mathematics 218; 7(2): ; N 7 = ; 7 = = ; } 7 = z }.~ (27) Thus, solutions of (26) can be easil found. The starting sstem K & for the general quintic problem can be obtained b putting these polnomial equations together block-wise. Similarl, we construct the following smmetric homotop for our problem with quintic nonlinearit where K & is defined as follows; 3 & ;,;,,; &,J = 1 JK & ;,;,,; & +JF & ;,;,,; &, (28) K & ;,;,,; & C?w z t 4.4. The Additional Smmetr for Odd Quintic Nonlinearit Similarl, the additional smmetr in the solutions of (1) can be obtained when ; = ~. That means if & is a solution of (1), then ±.±. & are also a solution besides &, where. = 1, which is named Z 7 smmetr. Therefore, we denote V Z 7 to the smmetr group of the discretized solutions and the homotop solution paths. C?w z ~ t =,. = 1,2,, Eample 1 Consider the following variable-coefficient fourth-order ODE with cubic polnomial nonlinearit + = + [ [,,1 = 1 = = 1 =, (29) 5. Numerical Results In this section, we present some numerical eamples to demonstrate the efficienc of smmetric homotop method. As we mentioned before, there are several software packages which can be used for tracking the representative paths of our smmetric homotop such as PHCpack, HOM4PS-2., and Bertini. Hence, in this paper, the numerical eperiments were performed b using PHCpack which has some properties such as accepting start sstem defined b the user. The efficienc of our smmetric homotop is verified b comparison with the efficienc of the total degree homotop. Total degree homotop where = = m n and = [ = m n7 are chosen to be smmetric about the center of domain Ω. The Tables 1-4 show the details of solution for the discretized problem, which are obtained b using the total degree homotop and smmetric homotop. No. Sols refer to the numbers of solutions (including comple solutions) which increase eponentiall with the numbers of eigenfunctions basis 9. No. Real Sols refer to the numbers of real solutions which are obtained b solving the target sstem of polnomial equations b the total degree homotop. No. Rep-Sols and No. Real Rep-Sols refer to the numbers of representative solutions and the numbers of real representative solutions obtained b using smmetric homotop, respectivel. Note that, such BVPs (29) ma have infinitel man solutions [23], and the discretization problem has onl finite solutions. Table 1. The data of discretized problem (29) with cubic polnomial nonlinearit in 1. Smmetric homotop N No. Sols No. Real Sols Time No. Rep-Sols No. Rep-Real Sols Time ms 6 4 ms ms ms ms ms s626ms s278ms s222ms s454ms m12s64ms m35s33ms m24s148ms m33s347ms h42m52s372ms m24s928ms h 4m51s161ms h12m18s133ms The details of solution for discretized problem (29) are listed in Table 1. When 9 = 1, we observe that the number of solutions for total degree homotop is approimatel equal to 2 times the number of representative solutions, and the epected time b total degree homotop is approimatel equal to 2 times the epected time b smmetric homotop Eample 2 Consider the following variable-coefficient fourth-order ODE with an odd cubic polnomial nonlinearit + = [, Ω,1

7 64 Abdrhaman Mahmoud et al.: Solving Variable-Coefficient Fourth-Order ODEs with Polnomial Nonlinearit b Smmetric Homotop Method = 1 = = 1 =, (3) where = m n and = m n7 are chosen to be smmetric about the center of domain Ω. Table 2 shows the details of solution for discretized problem (3). When 9 = 1, the number of solutions for total degree homotop is approimatel equal to 4 times the number of representative solutions, and the epected time b total degree homotop is approimatel equal to 4 times the epected time b smmetric homotop. Due to the additional smmetr in the solution set of (3), we epect the efficienc of smmetric homotop is more, comparing to solution of the problem with general cubic polnomial obtained in Table 1. Total degree homotop Table 2. The data of discretized problem (3) with an odd cubic polnomial nonlinearit in 1. Smmetric homotop N No. Sols No. Real Sols Time No. Rep-Sols No. Rep-Real Sols Time ms 4 3 ms ms ms ms ms s46ms ms s672ms s516ms s256ms s236ms m5s672ms m15s11ms m53s536ms m24s179ms h15m43s774ms m21s779ms As previousl mentioned, the number of solutions increases with the number of eigenfunctions basis 9. The solution set ma contain spurious solutions, which means that approimation solution is not closed to original solutions for ODEs. Therefore, removing spurious solutions with certain filters will be necessar. The filtering strateg used in [11] can be a possible wa to remove spurious solutions. The basic idea of this filter is based on error estimates of EEM, i.e., & N,9 f, &,9 f X, where & 1 &, and, is a generic positive constant independent of 9. The approimate solutions & of discretized problems on successivel finer levels satisf the error estimates, and then can be viewed as a solution path ; 9 parameterized b discretization level 9. Thus, the true approimate solutions should lie on a solution path. When 9 is large, the Cauch's criterion for convergence implies that & &X is ver small. Appling Newton's method with initial guesses & = &f to the discretization sstem, one can epect convergence for nonspurious solutions. The representative solutions for discretized problem (3) after filtering with 9 = 2 are displaed in Figure

8 Applied and Computational Mathematics 218; 7(2):

9 66 Abdrhaman Mahmoud et al.: Solving Variable-Coefficient Fourth-Order ODEs with Polnomial Nonlinearit b Smmetric Homotop Method Figure 1. The representative solutions for (3) after filtering with 9 = 2.

10 Applied and Computational Mathematics 218; 7(2): Eample 3 Consider the following variable-coefficient fourth-order ODE with quintic polnomial nonlinearit + = + [ [ ~ ~, Ω,1 = 1 = = 1 =, (31) where = = 7 = m n and = [ = ~ = m n7 are chosen to be smmetric about the center of domain Ω. The details of solution for discretized problem (31) are listed in Table 3. Total degree homotop Table 3. The data of discretized problem (31) with quintic polnomial nonlinearit in 1. Smmetric homotop N No. Sols No. Real Sols Time No. Rep-Sols No. Rep-Real Sols Time ms ms ms ms s892ms s547ms m29s435ms m15s3ms h 2m 7s489ms m52s373ms h59m15s81ms h51m5s51ms 5.4. Eample 4 Consider the following variable-coefficient fourth-order ODE with an odd quintic polnomial nonlinearit + = ~, Ω,1 = 1 = = 1 =, (32) where = m n and = m n7 are chosen to be smmetric about the center of domain Ω. Table 4. The data of discretized problem (32) with an odd quintic polnomial nonlinearit in 1. Total degree homotop Smmetric homotop N No. Sols No. Real Sols Time No. Rep-Sols No. Rep-Real Sols Time ms 11 3 ms ms ms s344ms s234ms m41s23m s553ms m14s611ms m 5s 8ms h46m42s993ms h24m4s948ms Table 4 shows the details of solution for discretized problem (32). When 9 = 7, the number of solutions for total degree homotop is approimatel equal to 2 times the number of representative solutions, and the epected time b total degree homotop is approimatel equal to 2 times the epected time b smmetric homotop. Similarl, because of the additional smmetr in the solution set of (32), the efficienc of smmetric homotop is more, comparing to solution of the problem with general quintic polnomial obtained in Table 3. The representative solutions for discretized problem (32) after filtering with 9 = 14 are displaed in Figure 2.

11 68 Abdrhaman Mahmoud et al.: Solving Variable-Coefficient Fourth-Order ODEs with Polnomial Nonlinearit b Smmetric Homotop Method

12 Applied and Computational Mathematics 218; 7(2): Figure 2. The representative solutions for (32) after filtering with 9 = 14.

13 7 Abdrhaman Mahmoud et al.: Solving Variable-Coefficient Fourth-Order ODEs with Polnomial Nonlinearit b Smmetric Homotop Method 6. Conclusion In this paper, a numerical technique based on EEM is presented to obtain the approimate solutions for variablecoefficient fourth-order ODEs with (cubic/quintic) polnomial nonlinearit. We have etensivel used the smmetr with dihedral group V which leads to simple computations. The smmetric homotop constructed for solving discretization sstems of ODEs can preserve the smmetr and reduce the computational cost. Numerical results demonstrated that the smmetric homotop method is efficient and promising. Acknowledgements This work is supported in part b the National Natural Science Foundation of China ( , ) and in part b the Fundamental Research Funds for the Central Universities (DUT16LK36). References [1] Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations, Journal of Mathematical Analsis and Applications, 27 (22), pp [2] Y. Yang, Fourth-order two-point boundar value problems, Proceedings of the American Mathematical Societ, (1988), pp [3] G. Bonanno and B. Di Bella, A boundar value problem for fourth-order elastic beam equations, Journal of Mathematical Analsis and Applications, 343 (28), pp [4] M. do Rosário Grossinho, L. Sanchez, and S. A. Tersian, On the solvabilit of a boundar value problem for a fourth-order ordinar differential equation, Applied Mathematics Letters, 18 (25), pp [5] L. Greenberg and M. Marletta, Numerical methods for higher order sturm-liouville problems, Journal of Computational and Applied Mathematics, 125 (2), pp [6] Z. S. Aliev and F. M. Namazov, Spectral properties of a fourth-order eigenvalue problem with spectral parameter in the boundar conditions, Electronic Journal of Differential Equations, 217 (217), pp [7] R. P. Agarwal, Boundar value problems for higher order differential equations, tech. report, [8] G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Non-linear Analsis: Theor, Methods &Applications, 68 (28), pp [9] X. L. Liu and W. T. Li, Eistence and multiplicit of solutions for fourth-order boundar value problems with three parameters, Mathematical and Computer Modelling, 46 (27), pp [1] A. Cabada, R. Precup, L. Saavedra, and S. A. Tersian, Multiple positive solutions to a fourth-order boundar-value problem, Electronic Journal of Differential Equations, 216 (216), pp [11] X. Zhang, J. Zhang, and B. Yu, Eigenfunction epansion method for multiple solutions of semilinear elliptic equations with polnomial nonlinearit, SIAM Journal on Numerical Analsis, 51 (213), pp [12] R. L. Burden and J. D. Faires, Numerical analsis, Cengage Learning, 211. [13] J. Aleander and J. A. Yorke, The homotop continuation method: numericall implementable topological procedures, Transactions of the American Mathematical Societ, 242 (1978), pp [14] T. Y. Li, Numerical solution of multivariate polnomial sstems b homotop continuation methods, Acta numerica, 6 (1997), pp [15] T. Y. Li, Numerical solution of polnomial sstems b homotop continuation methods, Handbook of numerical analsis, 11 (23), pp [16] J. Verschelde, Algorithm 795: Phcpack: A general-purpose solver for polnomial sstems b homotop continuation, ACM Transactions on Mathematical Software (TOMS), 25 (1999), pp [17] T. L. Lee, T. Y. Li, and C. H. Tsai, Hom4ps-2.: a software package for solving polnomial sstems b the polhedral homotop continuation method, Computing, 83 (28), pp [18] D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler, Bertini: Software for numerical algebraic geometr (26), Software available at nd. edu. [19] A. J. Sommese and C. W. Wampler II, The Numerical solution of sstems of polnomials arising in engineering and science, World Scientific, 25. [2] E. L. Allgower, D. J. Bates, A. J. Sommese, and C. W. Wampler, Solution of polnomial sstems derived from differential equations, Computing, 76 (26), pp [21] X. Zhang, J. Zhang, and B. Yu, Smmetric homotop method for discretized elliptic equations with cubic and quantic nonlinearities, Journal of Scientific Computing, 7 (217), pp [22] W. Hao, J. D. Hauenstein, B. Hu, and A. J. Sommese, A bootstrapping approach for computing multiple solutions of differential equations, Journal of Computational and Applied Mathematics, 258 (214), pp [23] S. M. Khalkhali, S. Heidarkhani, and A. Razani, Infinitel man solutions for a fourth-order boundar-value problem, Electronic Journal of Differential Equations, 212 (212), pp