Skip to main content
SpringerLink
Log in

Burger and Seven-Order KdV Equations Using Modified Differential Transform Method

  • Short Communication
  • Published:
National Academy Science Letters Aims and scope Submit manuscript

Abstract

In the present paper, a modified differential transform method (DTM) and DTM with finite difference method (FDM) has been given to solve Burger and seven-order KdV equations. Traditionally, DTM is a tool to solve differential and integral equations. However, Lax–Sawada–Kotera–Ito seven-order KdV equations are usually very complicated, so the calculation of the numerical solution becomes difficult. On the other hand, FDM is very easy for coding. The present study combines ADM and FDM which is an easy method with high accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

References

  1. Kappeler T, Schaad B, Topalov P (2013) Qualitative features of periodic solutions of KdV. Commun Partial Differ Equ 38:1626–1673

    Article  MathSciNet  MATH  Google Scholar 

  2. Martel Y, Merl F (2003) Qualitative results on the generalized critical KdV equation. Lectures on partial differential equations. New studies advanced mathematics, vol 2. International Press, Somerville, MA, pp 175–179

    Google Scholar 

  3. Belashov VYu, Tyunina SG (1997) Qualitative analysis and the asymptotic behavior of solutions of generalized equations of the KdV class. Izv Vyssh Uchebn Zaved Radiofiz 40:328–344

    MathSciNet  Google Scholar 

  4. Li Wen Shen (1994) The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation. Appl Math Mech 15:491–498

    Article  MathSciNet  MATH  Google Scholar 

  5. Guo Bo Ling (1983) Existence and uniqueness of the global solution of the Cauchy problem and the periodic initial value problem for a class of coupled systems of KdV-nonlinear Schrodinger equations. Acta Math Sinica 26:513–532

    MathSciNet  MATH  Google Scholar 

  6. Jokhadze O (2008) On existence and nonexistence of global solutions of Cauchy–Goursat problem for nonlinear wave equations. J Math Anal Appl 340:1033–1045

    Article  MathSciNet  MATH  Google Scholar 

  7. Amorim P, Figueira M (2013) Convergence of a numerical scheme for a coupled Schrodinger-KdV system. Rev Mat Complut 26:409–426

    Article  MathSciNet  MATH  Google Scholar 

  8. Trogdon T, Deconinck B (2014) A numerical dressing method for the nonlinear superposition of solutions of the KdV equation. Nonlinearity 27:67–86

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Saha Ray S (2013) Numerical solutions and solitary wave solutions of fractional KDV equations using modified fractional reduced differential transform method. Comput Math Math Phys 53:1870–1881

    Article  MathSciNet  Google Scholar 

  10. Idczak D, Majewski M, Walczak S (2003) Stability analysis of solutions to an optimal control problem associated with a Goursat–Darboux problem, Multidimensional systems and iterative learning control. Int J Appl Math Comput 13:29–44

    MathSciNet  MATH  Google Scholar 

  11. Drignei M (2013) A numerical method for solving a Goursat–Cauchy boundary value problem. Appl Math Comput 220:123–141

    Article  MathSciNet  MATH  Google Scholar 

  12. Al-Mazmumy MA (2011) Adomian decomposition method for solving Goursat’s problems. Appl Math (Irvine) 2:975–980

    Article  MathSciNet  Google Scholar 

  13. Wazwaz A (1995) The decomposition method for approximate solution of the Goursat problem. Appl Math Comput 69:299–311

    Article  MathSciNet  MATH  Google Scholar 

  14. Puhov GE (1976) The application of Taylor transforms to the solution of differential equations. Elektronikai Modelirovanie 128:18–23

    MathSciNet  Google Scholar 

  15. Kurnaz A, Oturanc G (2005) The differential transform approximation for the system of ordinary differential equations. Int Comput Math 82:709–719

    Article  MathSciNet  MATH  Google Scholar 

  16. Mirzaee F (2011) Differential transform method for solving linear and nonlinear systems of ordinary differential equations. Appl Math Sci 70:3465–3472

    MathSciNet  MATH  Google Scholar 

  17. Chen CK, Ho SH (1999) Solving partial differential equations by two-dimensional differential transform method. Appl Math Comput 106:171–179

    Article  MathSciNet  MATH  Google Scholar 

  18. Jang MJ, Chen CL, Liu YC (2001) Two-dimensional differential transform for partial differential equations. Appl Math Comput 121:261–270

    Article  MathSciNet  MATH  Google Scholar 

  19. Etinkaya AC, Kıymaz O, Amlı JC (2011) Solutions of nonlinear PDE’s of fractional order with generalized differential transform method. Int Math Forum 6:39–47

    MathSciNet  Google Scholar 

  20. Liu H, Song Y (2007) Differential transform method applied to high index differential-algebraic equations. Appl Math Comput 184:748–753

    Article  MathSciNet  MATH  Google Scholar 

  21. Ayaz F (2004) Applications of differential transform method to differential-algebraic equations. Appl Math Comput 152:649–657

    Article  MathSciNet  MATH  Google Scholar 

  22. Zou L, Wang Z, Zong Z (2009) Generalized differential transform method to differential-difference equation. Phys Lett A 373:4142–4151

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  23. Arikoglu A, Ozkol I (2006) Solution of differential-difference equations by using differential transform method. Appl Math Comput 181:153–162

    Article  MathSciNet  MATH  Google Scholar 

  24. Tari A, Shahmorad S (2011) Differential transform method for the system of two-dimensional nonlinear Volterra integro differential equations. Comput Math Appl 61:2621–2629

    Article  MathSciNet  MATH  Google Scholar 

  25. Nazari D, Shahmorad S (2010) Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions. J Comput Appl Math 234:883–891

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu J, Hou G (2011) Numerical solutions of the space- and time fractional coupled Burgers equations by generalized differential transform method. Appl Math Comput 217:7001–7008

    Article  MathSciNet  MATH  Google Scholar 

  27. Alquran MT, Al-Khaled K (2010) Approximations of Sturm–Liouville eigenvalues using sinc-Galerkin and differential transform methods. Appl Appl Math 5:128–147

    MathSciNet  MATH  Google Scholar 

  28. Abassy TA, El-Tawil MA, Saleh HK (2007) The solution of Burgers’ and good Boussinesq equations using ADM-Pade technique. Chaos Solitons Fractals 32:1008–1026

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Salas AH, Gomez SC, Acevedo FB (2010) Computing exact solutions to a generalized Lax–Sawada–Kotera–Ito seventh-order KdV equation. Math Probl Eng 2010:1–7. doi:10.1155/2010/524567

    MathSciNet  MATH  Google Scholar 

  30. Lin YW, Lu TT, Chen CK (2013) Adomian decomposition method using integrating factor. Commun Theor Phys 60:159–164

    Article  MathSciNet  MATH  Google Scholar 

  31. Singh N, Kumar M (2013) Adomian decomposition method for computing eigen-values of singular Sturm–Liouville problems. Nat Acad Sci Lett 36:311–318

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, National Cheng Kung University, Tainan, 70101, Taiwan, ROC

    Yinwei Lin & Cha’o-Kuang Chen

Authors
  1. Yinwei Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cha’o-Kuang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yinwei Lin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, Y., Chen, CK. Burger and Seven-Order KdV Equations Using Modified Differential Transform Method. Natl. Acad. Sci. Lett. 39, 353–357 (2016). https://doi.org/10.1007/s40009-016-0462-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40009-016-0462-0

Keywords

Search

Navigation