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Optimal solution of a general class of nonlinear system of fractional partial differential equations using hybrid functions

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Abstract

This paper introduces a general class of nonlinear system of fractional partial differential equations with initial and boundary conditions. A hybrid method based on the transcendental Bernstein series and the generalized shifted Chebyshev polynomials is proposed for finding the optimal solution of the nonlinear system of fractional partial differential equations. The solution of the nonlinear system of fractional partial differential equations is expanded in terms of the transcendental Bernstein series and the generalized shifted Chebyshev polynomials, as basis functions with unknown free coefficients and control parameters. The corresponding operational matrices of fractional derivatives are then derived for the basis functions. These basis functions, with their operational matrices of fractional order derivatives and the Lagrange multipliers, transform the problem into a nonlinear system of algebraic equations. By means of Darbo’s fixed point theorem and Banach contraction principle, an existence result and a unique result for the solution of the nonlinear system of fractional partial differential equations are obtained, respectively. The convergence analysis is discussed and several illustrative experiments illustrate the efficiency and accuracy of the proposed method.

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References

  1. Hesameddini E, Shahbazi M (2019) Solving multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type using Bernstein polynomials method. Appl Numer Math 136:122–138

    MathSciNet  MATH  Google Scholar 

  2. Yüzbasi S (2016) A collocation method based on Bernstein polynomials to solve nonlinear Fredholm–Volterra integro-differential equations. Appl Math Comput 273(15):142–154

    MathSciNet  MATH  Google Scholar 

  3. Wang J, Xu TZ, Wang GW (2018) Numerical algorithm for time-fractional Sawada–Kotera equation and Ito equation with Bernstein polynomials. Appl Math Comput 338:1–11

    MathSciNet  MATH  Google Scholar 

  4. Asgari M, Ezzati R (2017) Using operational matrix of two-dimensional Bernstein polynomials for solving two-dimensional integral equations of fractional order. Appl Math Comput 307:290–298

    MathSciNet  MATH  Google Scholar 

  5. Javadi Sh, Babolian E, Taheri Z (2016) Solving generalized pantograph equations by shifted orthonormal bernstein polynomials. J Comput Appl Math 303:1–14

    MathSciNet  MATH  Google Scholar 

  6. Safaie E, Farahi MH, Farmani Ardehaie M (2015) An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials. Comput Appl Math 34(3):831–846

    MathSciNet  MATH  Google Scholar 

  7. Chen Y, Yi M, Chen C, Yu C (2012) Bernstein polynomials method for fractional convection–diffusion equation with variable coefficients. CMES Comput Model Eng Sci 83:639–654

    MathSciNet  MATH  Google Scholar 

  8. Behiry SH (2014) Solution of nonlinear Fredholm integro-differential equations using a hybrid of block pulse functions and normalized Bernstein polynomials. J Comput Appl Math 260:258–265

    MathSciNet  MATH  Google Scholar 

  9. Vijender N (2019) Bernstein fractal trigonometric approximation. Acta Appl Math 159(1):11–27

    MathSciNet  MATH  Google Scholar 

  10. Wang H, Zhang L (2018) Jacobi polynomials on the Bernstein ellipse. J Sci Comput 75:457–477

    MathSciNet  MATH  Google Scholar 

  11. Zi-Qiang B, Yan G, Chia-Ming F (2019) A direct Chebyshev collocation method for the numerical solutions of three-dimensional Helmholtz-type equations. Eng Anal Bound Elem 104:26–33

    MathSciNet  MATH  Google Scholar 

  12. Khatri Ghimire B, Tian HY, Lamichhane AR (2016) Numerical solutions of elliptic partial differential equations using Chebyshev polynomials. Comput Math Appl 72(4):1042–1054

    MathSciNet  MATH  Google Scholar 

  13. Yang C (2018) Modified Chebyshev collocation method for pantograph-type differential equations. Appl Numer Math 134:132–144

    MathSciNet  MATH  Google Scholar 

  14. Doha EH, Abdelkawy MA, Amin AZM, Lopes AM (2018) A space-time spectral approximation for solving nonlinear variable-order fractional sine and Klein–Gordon differential equations. Comput Appl Math 37(5):6212–6229

    MathSciNet  MATH  Google Scholar 

  15. Ezz-Eldien SS, Wang Y, Abdelkawy MA, Zaky MA, Aldraiweesh AA, Tenreiro Machado J (2020) Chebyshev spectral methods for multi-order fractional neutral pantograph equations. Nonlinear Dynam 100:3785–3797

    Google Scholar 

  16. Pfeiffer BM, Marquardt W (1996) Symbolic semi-discretization of partial differential equation systems. Math Comput Simulat 42:617–628

    MathSciNet  MATH  Google Scholar 

  17. Sun ZZ, Xu X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math 56:193–209

    MathSciNet  MATH  Google Scholar 

  18. Gimena L, Gonzaga P, Gimena FN (2014) Boundary equations in the finite transfer method for solving differential equation systems. Appl Math Model 38:2648–2660

    MathSciNet  MATH  Google Scholar 

  19. Singla K, Gupta RK (2016) On invariant analysis of some time fractional nonlinear systems of partial differential equations. I. J Math Phys 57(10):101504. https://doi.org/10.1063/1.4964937

    Article  MathSciNet  MATH  Google Scholar 

  20. Abdel-Halim Hassan IH (2008) Application to differential transformation method for solving systems of differential equations. Appl Math Model 32:2552–2559

    MathSciNet  MATH  Google Scholar 

  21. Alliera CHD, Amster P (2018) Systems of delay differential equations: analysis of a model with feedback. Commun Nonlinear Sci Numer Simulat 65:299–308

    MathSciNet  MATH  Google Scholar 

  22. Ablinger J, Blümlein J, Marquard P, Rana N, Schneider C (2019) Automated solution of first order factorizable systems of differential equations in one variable. Nucl Phys B 939:253–291

    MathSciNet  MATH  Google Scholar 

  23. Ball JM (1983) systems of nonlinear partial differential equations. Springer, Berlin

    MATH  Google Scholar 

  24. Zaky MA (2020) Nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl Numer Math 154:205–222

    MathSciNet  MATH  Google Scholar 

  25. Leon Pritchard F (2003) On implicit systems of differential equations. J Differ Equ 194:328–363

    MathSciNet  MATH  Google Scholar 

  26. Feng TF, Chang CH, Chen JB, Zhang HB (2019) The system of partial differential equations for the \(c_0\) function. Nucl Phys B 940:130–189

    MATH  Google Scholar 

  27. Jaros J, Kusano T (2014) On strongly monotone solutions of a class of cyclic systems of nonlinear differential equations. J Math Anal Appl 417:996–1017

    MathSciNet  MATH  Google Scholar 

  28. Guaily AG, Epstein M (2013) Boundary conditions for hyperbolic systems of partial differentials equations. J Adv Res 4(4):321–329

    Google Scholar 

  29. Filbet F, Xiong T (2018) A hybrid discontinuous Galerkin scheme for multi-scale kinetic equations. J Comput Phys 372(1):841–863

    MathSciNet  MATH  Google Scholar 

  30. Menci M, Papi M (2019) Global solutions for a path-dependent hybrid system of differential equations under parabolic signal. Nonlinear Anal 184:172–192

    MathSciNet  MATH  Google Scholar 

  31. Feib C, Shen M, Fei W, Mao X, Yan L (2019) Stability of highly nonlinear hybrid stochastic integro-differential delay equations. Nonlinear Anal Hybri 31:180–199

    MathSciNet  MATH  Google Scholar 

  32. Dehao R, Jiaowan XL (2019) Stability of hybrid stochastic functional differential equations. Appl Math Comput 346(1):832–841

    MathSciNet  MATH  Google Scholar 

  33. Chen C, Zhang X, Liu Z (2020) A high-order compact finite difference scheme and precise integration method based on modified Hopf–Cole transformation for numerical simulation of n-dimensional Burgers’ system. Appl Math Comput 372:125009

    MathSciNet  MATH  Google Scholar 

  34. Zaky MA (2020) An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl Numer Math 154:205–222

    MathSciNet  MATH  Google Scholar 

  35. 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

    MathSciNet  MATH  Google Scholar 

  36. Heydari MH, Avazzadeh Z (2020) Numerical study of non-singular variable-order time fractional coupled Burgers’ equations by using the Hahn polynomials. Eng Comput. https://doi.org/10.1007/s00366-020-01036-5

  37. Sabir M, Shah A, Muhammad W, Ali I, Bastian P (2017) A mathematical model of tumor hypoxia targeting in cancer treatment and its numerical simulation. Comput Math Appl 74(12):3250–3259

    MathSciNet  MATH  Google Scholar 

  38. Hendy AS, Zaky MA (2020) Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations. Eng Comput. https://doi.org/10.1007/s00366-020-01095-8

  39. Jafari H, Seifi S (2009) Solving a system of nonlinear fractional partial differential equations using homotopy analysis method. Commun Nonlinear Sci Numer Simul 14:1962–1969

    MathSciNet  MATH  Google Scholar 

  40. Kreyszig E (1978) Introductory functional analysis with applications. Wiley, Hoboken

    MATH  Google Scholar 

  41. Seemab A, Remhman M (2020) Existence of solution of an infinite system of generalized fractional differential equations by Darbo’s fixed point theorem. J Comput Appl Math 364:112355

    MathSciNet  MATH  Google Scholar 

  42. Sun S, Li Q, Li Y (2011) Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations. Comput Math Appl 64:3310–3320

    MathSciNet  MATH  Google Scholar 

  43. Kuratowski K (1939) Sur les espaces complets. Fund Math 15:301–309

    MATH  Google Scholar 

  44. Banas J, Goebel K (1980) Measures of Noncompactness in Banach Spaces, vol 60. Lecture Notes in Pure and Applied Mathematics. Dekker, NewYork

  45. Roudin W (1976) Principles of mathematical analysis, 3rd edn. McGraw-Hill. Inc, New York

    Google Scholar 

  46. Gasea M, Sauer T (2000) On the history of multivariate polynomial interpolation. J Comput Appl Math 122:23–35

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Anand International College of Engineering, Jaipur, 303012, India

    H. Hassani

  2. Department of Electrical Engineering, Polytechnic of Porto, Institute of Engineering, R. Dr. António Bernardino de Almeida 431, Porto, 4249-015, Portugal

    J. A. Tenreiro Machado

  3. Department of Mathematics, Yasouj University, Yasouj, Iran

    E. Naraghirad

  4. Department of Applied Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China

    Z. Avazzadeh

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  1. H. Hassani
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  2. J. A. Tenreiro Machado
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  3. E. Naraghirad
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  4. Z. Avazzadeh
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Correspondence to E. Naraghirad.

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Hassani, H., Machado, J.A.T., Naraghirad, E. et al. Optimal solution of a general class of nonlinear system of fractional partial differential equations using hybrid functions. Engineering with Computers 39, 2401–2431 (2023). https://doi.org/10.1007/s00366-022-01627-4

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  • DOI: https://doi.org/10.1007/s00366-022-01627-4

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