Skip to main content
SpringerLink
Log in

Genocchi Wavelet Method for the Solution of Time-Fractional Telegraph Equations with Dirichlet Boundary Conditions

  • Research Paper
  • Published:
Iranian Journal of Science Aims and scope Submit manuscript

Abstract

The present paper suggests a novel, efficient operational matrix technique on the basis of block-pulse functions and Genocchi wavelets to solve time-fractional telegraph equations considering Dirichlet boundary conditions. First, a brief overview of the Genocchi polynomials, corresponding wavelets, and fundamental characteristics is presented. Then, the same functions and their suitable characteristics are employed to formulate the Genocchi wavelet-like operational matrices of fractional integration. Using the suggested technique, the fractional model is reduced into a system of algebraic equations, which is solvable by employing the classical Newton’s iteration technique. A comparison is made between the estimated solutions of the time-fractional telegraph equation and the present approaches, such as the Legendre wavelet and the Fibonacci wavelet method. According to the numerical results, accurate results are obtained using the Genocchi method, and therefore, it is computationally more effective compared to the present approaches.

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
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Abbasa S, Benchohra M (2014) Fractional order integral equations of two independent variables. Appl Math Comput 227:755–761

    MathSciNet  Google Scholar 

  • Abdollahy Z, Mahmoudi Y, Salimi Shamloo A, Baghmisheh M (2022) Haar wavelets method for time fractional Riesz space telegraph equation with separable solution. Rep Math Phys 37:81–96

    Article  MathSciNet  Google Scholar 

  • Abdulazeez ST, Modanli M (2022) Solutions of fractional order pseudo-hyperbolic telegraph partial differential equations using finite difference method. Alex Eng J 63:12443–12451

    Article  Google Scholar 

  • Adams E, Spreuer H (1975) Uniqueness and stability for boundary value problems with weakly coupled systems of nonlinear integro-differential equations and application to chemical reactions. J Math Anal Appl 49:393–410

    Article  MathSciNet  Google Scholar 

  • Aghazadeh N, Khajehnasiri AA (2013) Solving nonlinear two-dimensional Volterra integro-differential equations by block-pulse functions. Math Sci 7:1–6

    Article  MathSciNet  Google Scholar 

  • Cinar M, Secer A, Bayram M (2021) An application of Genocchi wavelets for solving the fractional Rosenau–Hyman equation. Alex Eng J 60:5331–5340

    Article  Google Scholar 

  • Dehestani H, Ordokhani Y, Razzaghi M (2019) On the applicability of Genocchi wavelet method for different kinds of fractional-order differential equations with delay. Numer Linear Algebra Appl 10:1–12

    MathSciNet  Google Scholar 

  • Dehestani H, Ordokhani Y, Razzaghi M (2020) The novel operational matrices based on 2D-Genocchi polynomials: solving a general class of variable-order fractional partial integro-differential equations. Comput Appl Math 259:1–32

    MathSciNet  Google Scholar 

  • Doha EH, Bhrawy AH, Ezz-Eldien SS (2012) A new Jacobi operational matrix: an application for solving fractional differential equations. Appl Math Model 36:4931–4943

    Article  MathSciNet  Google Scholar 

  • El-Gamel M, Mohamed N, Adel W (2022a) Numerical study of a nonlinear high order boundary value problems using Genocchi collocation technique. Int J Appl Comput Math 143:1–18

    MathSciNet  Google Scholar 

  • El-Gamel M, Adel W, El-Azab MS (2022b) Eigenvalues and eigenfunctions of fourth-order Sturm–Liouville problems using Bernoulli series with Chebychev collocation points. Int J Appl Comput Math 16:97–104

    MathSciNet  Google Scholar 

  • Gaul L, Klein P, Kempfle S (1991) Damping description involving fractional operators. Mech Syst Signal Process 5:81–88

    Article  Google Scholar 

  • Hashemizadeh E, Ebadi MA, Noeiaghdam S (2020) Matrix method by Genocchi polynomials for solving nonlinear volterra integral equations with weakly singular kernels. Symmetry 176:1–16

    Google Scholar 

  • Hashmi MS, Aslam U, Singh J, Nisar KS (2022) An efficient numerical scheme for fractional model of telegraph equation. Alex Eng J 61:6383–9393

    Article  Google Scholar 

  • Hesameddini E, Shahbazi M (2018) Hybrid Bernstein Block-Pulse functions for solving system of fractional integro-differential equations. J Comput Appl Math 95:644–651

    MathSciNet  Google Scholar 

  • Heydari MH, Hooshmandasla MR, Mohammadi F, Cattani C (2014) Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun Nonlinear Sci Numer Simul 19:37–48

    Article  MathSciNet  Google Scholar 

  • Hosseininia M, Heydari MH (2019) Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag-Leffler non-singular kernel. Chaos, Solitons & Fractals 127:389–399

    Article  MathSciNet  Google Scholar 

  • Hwang C, Shih YP (1982) Parameter identification via Laguerre polynomials. Int J Syst Sci 13:209–17

    Article  MathSciNet  Google Scholar 

  • Isah A, Phang C (2016) Genocchi wavelet-like operational matrix and its application for solving non-linear fractional differential equations. Open Phys 14:463–472

    Article  Google Scholar 

  • Isah A, Phang C (2017) New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials. J King Saud Univ Sci 14:1–7

    Google Scholar 

  • Ismaeelpour T, Hemmat AA, Saeedi H (2017) B-spline operational matrix of fractional integration. Optik Int J Light Electron Opt 130:291–305

    Article  Google Scholar 

  • Kanwal A, Phang C, Iqbal U (2018) Numerical solution of fractional diffusion wave equation and fractional Klein–Gordon equation via two-dimensional Genocchi polynomials with a Ritz–Galerkin method. Computation 16:1–12

    Google Scholar 

  • Khajehnasiri AA, Ebadian A (2023) Genocchi operational matrix method and their applications for solving fractional weakly singular two-dimensional partial Volterra integral equation. U.P.B. Sci Bull Ser A 85:1–18

    MathSciNet  Google Scholar 

  • Khajehnasiri AA, Ezzati R (2021) Boubaker polynomials and their applications for solving fractional two-dimensional nonlinear partial integro-differential Volterra integral equations. Comput Appl Math 41:1–18

    MathSciNet  Google Scholar 

  • Khajehnasiri AA, Safavi M (2021) Solving fractional Black-Scholes equation by using Boubaker functions. Math Methods Appl Sci 44:8505–8515

    Article  MathSciNet  Google Scholar 

  • Khajehnasiri AA, Afshar ER, Kermani M (2021) Solving fractional two-dimensional nonlinear partial Volterra integral equation by using Bernoulli wavelet. Iran J Sci Technol Trans Sci 16:983–995

    Article  MathSciNet  Google Scholar 

  • Kumar S (2014) A new analytical modelling for fractional telegraph equation via Laplace transform. Appl Math Model 38:3154–3163

    Article  MathSciNet  Google Scholar 

  • LiGong X, Liu XH, Xiong X (2019) Chaotic analysis and adaptive synchronization for a class of fractional order financial system. Phys A 522(15):33–42

    MathSciNet  Google Scholar 

  • Mokhtary P (2017) Numerical analysis of an operational Jacobi Tau method for fractional weakly singular integro-differential equations. Appl Numer Math 121:52–67

    Article  MathSciNet  Google Scholar 

  • Mollahasani N, Afrooz Moghadam MMK (2016) A new treatment based on hybrid functions to the solution of telegraph equations of fractional order. Appl Math Model 40:2804–2814

    Article  MathSciNet  Google Scholar 

  • Nemati S, Sedaghat S, Mohammadi I (2016) A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels. J Comput Appl Math 308:231–242

    Article  MathSciNet  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    Google Scholar 

  • Rabiei K, Ordokhani Y (2018) Boubaker hybrid functions and their application to solve fractional optimal control and fractional variational problems. Appl Math 5:541–567

    Article  MathSciNet  Google Scholar 

  • Rahmani Fazli H, Hassani F, Ebadian A, Khajehnasiri AA (2015) National economies in state-space of fractional-order financial system. Afr Mat 10:1–12

    Google Scholar 

  • Rawashdeh E (2006) Numerical solution of fractional integro-differential equations by collocation method. Appl Math Comput 176:1–6

    MathSciNet  Google Scholar 

  • Saadatmand A (2014) Bernstein operational matrix of fractional derivatives and its applications. Appl Math Model 38:1365–1372

    Article  MathSciNet  Google Scholar 

  • Safavi M, Khajehnasiri AA, Jafari A, Banar J (2021) A new approach to numerical solution of nonlinear partial mixed Volterra–Fredholm integral equations via two-dimensional triangular functions. Malays J Math Sci 15:489–507

    MathSciNet  Google Scholar 

  • Saha Ray S, Behera S (2020) Two-dimensional wavelets operational method for solving Volterra weakly singular partial integro-differential equations. J Comput Appl Math 366:1–29

    MathSciNet  Google Scholar 

  • Schiavane P, Constanda C, Mioduchowski A (2002) Integral methods in science and engineering. Birkhäuser, Boston

    Google Scholar 

  • Shah FA, Irfan M, Nisar KS, Matoog RT, Mahmoud EE (2021a) Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions. Res Phys 24:1–10

    Google Scholar 

  • Shah FA, Irfan M, Nisar KS, Matoog RT, Mahmoud EE (2021b) Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions. Res Phys 74:1–10

    Google Scholar 

  • Shekher Singh C, Singh Vineet H, Singh K, Singh OP (2016) Fractional order operational matrix methods for fractional singular integro-differential equation. Appl Math Model 40:10705–10718

    Article  MathSciNet  Google Scholar 

  • Singh U (2022) Numerical investigation of unconditionally stable spline function for three-dimensional time-fractional telegraph equations. Res Control Optim 40:1–17

    Google Scholar 

  • Singh PK, Saha Ray S (2022a) A novel operational matrix method based on Genocchi polynomials for solving n-dimensional stochastic Itâ–Volterra integral equation. Math Sci 17:1–11

    Google Scholar 

  • Singh PK, Saha Ray S (2022b) A novel operational matrix method based on Genocchi polynomials for solving n-dimensional stochastic Itâ–Volterra integral equation. Math Sci 22:1–11

    Google Scholar 

  • Singh CS, Singh H, Singh VK, Singh OP (2016) Fractional order operational matrix methods for fractional singular integro-differential equation. Appl Math Model 366:10705–10718

    Article  MathSciNet  Google Scholar 

  • Srivastava HM, Adel W, Izadi M, El-Sayed AA (2023) Solving some physics problems involving fractional-order differential equations with the Morgan–Voyce polynomials. Fractal Fract 7:1–19

    Article  Google Scholar 

  • Sweilam NH, Nagy AM, El-Sayed AA (2016) Solving time-fractional order telegraph equation via Sinc-Legendre collocation method. Mediterr J Math 13:5119–5133

    Article  MathSciNet  Google Scholar 

  • Xu X, Xu D (2018) Legendre wavelets direct method for the numerical solution of time-fractional order telegraph equations. Mediterr J Math 18:15–27

    MathSciNet  Google Scholar 

  • Yi M, Huang J, Wei J (2013) Block pulse operational matrix method for solving fractional partial differential equation. Appl Math Comput 221:121–131

    MathSciNet  Google Scholar 

  • Zhao Z, Li C (2012) Fractional difference/finite element approximations for the time-space fractional telegraph equation. Appl Math Comput 219:2975–2988

    MathSciNet  Google Scholar 

  • Zurigat M, Momani S, Alawneh A (2009) Homotoy analysis method for systems of fractional integro-differential equations. Neural Parallel Sci Comput 17:169–186

    MathSciNet  Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran

    A. A. Khajehnasiri & A. Ebadian

Authors
  1. A. A. Khajehnasiri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Ebadian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Khajehnasiri.

Ethics declarations

Conflict of interest

The author declare that there is no conflict of interest

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khajehnasiri, A.A., Ebadian, A. Genocchi Wavelet Method for the Solution of Time-Fractional Telegraph Equations with Dirichlet Boundary Conditions. Iran J Sci (2024). https://doi.org/10.1007/s40995-024-01635-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40995-024-01635-7

Keywords

Mathematics Subject Classification

Search

Navigation