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Exact solution with existence and uniqueness conditions for multi-dimensional time-space tempered fractional diffusion-wave equation

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Abstract

In this paper, we investigate the existence and uniqueness of the multi-dimensional time-space tempered fractional diffusion-wave (TSTFDW) equation by using the fixed point theory. We also found an exact solution of the multi-dimensional TSTFDW equation. The two-step Adomian decomposition method is proposed to achieve the exact solution of this problem. The main advantages of this method are its easy implementation and higher efficiency than other existing numerical methods.

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References

  1. Atangana A, Secer A (2013) A note on fractional order derivatives and table of fractional derivatives of some special functions. Abst Appl Anal. https://doi.org/10.1155/2013/279681

    Article  MathSciNet  MATH  Google Scholar 

  2. Ebaid A, Masaedeh B, El-Zahar E (2017) A new fractional model for the falling body problem. Chin Phys Lett. https://doi.org/10.1088/0256-307X/34/2/020201/pdf

    Article  Google Scholar 

  3. Ebaid A, El-Zahar ER, Aljohani AF, Salah B, Krid M, Machado JT (2019) Analysis of the two-dimensional fractional projectile motion in view of the experimental data. Nonlinear Dyn. https://doi.org/10.1007/s11071-019-05099-y

    Article  MATH  Google Scholar 

  4. Atangana A (2016) On the new fractional derivative and application to non-linear Fisher’s reaction-diffusion equation. Appl Math Comp 273:948–956

    Article  Google Scholar 

  5. Li Y, Chen Y, Podlubny I (2010) Stability of fractional-order non-linear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput Math Appl 59:1810–1821

    Article  MathSciNet  Google Scholar 

  6. Ahmad B, Sivasundaram S (2010) On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl Math Comput 217:480–487

    MathSciNet  MATH  Google Scholar 

  7. Doha EH, Bhrawyb AH, Ezz-Eldien SS (2011) A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput Math Appl 62:2364–2373

    Article  MathSciNet  Google Scholar 

  8. Kazem S, Abbasbandy S, Kumar S (2013) Fractional-order Legendre functions for solving fractional-order differential equations. App Math Mode 37:5498–5510

    Article  MathSciNet  Google Scholar 

  9. Wu GC (2011) A fractional variational iteration method for solving fractional nonlinear differential equations. Comput Math Appl 61:2186–2190

    Article  MathSciNet  Google Scholar 

  10. Jafari H, Seifi S (2009) Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation. Commun Nonlinear Sci Num Simul 14:2006–2012

    Article  MathSciNet  Google Scholar 

  11. Babolian E, Vahidi AR, Shoja A (2014) An efficient Method for non-linear fractional differential equations: combination of the Adomian decomposition method and spectral method. Ind J Pure Appl Math 45:1017–1028

    Article  Google Scholar 

  12. Odibat Z, Momani S, Xu H (2010) A reliable algorithm of homotopy analysis method for solving non-linear fractional differential equations. Appl Math Mod 34:593–600

    Article  Google Scholar 

  13. Hashima I, Abdulaziza O, Momani S (2009) Homotopy analysis method for fractional IVPs. Commun Nonlinear Sci Num Simul 14:674–684

    Article  MathSciNet  Google Scholar 

  14. KarimiVanani S, Aminataei A (2011) Tau approximate solution of fractional partial differential equations. Comput Math Appl 62:1075–1083

    Article  MathSciNet  Google Scholar 

  15. El-Wakil SA, Elhanbaly A, Abdou MA (2006) Adomian decomposition method for solving fractional nonlinear differential equations. Appl Math Comput 182:313–324

    MathSciNet  MATH  Google Scholar 

  16. Assari P (2019) On the numerical solution of two-dimensional integral equations using a meshless local discrete Galerkin scheme with error analysis. Eng Comput 35:893–916

    Article  Google Scholar 

  17. Hassani H, Avazzadeh Z, Tenreiro Machado JA (2019) Numerical approach for solving variable-order space-time fractional telegraph equation using transcendental Bernstein series. Eng Comput. https://doi.org/10.1007/s00366-019-00736-x

    Article  Google Scholar 

  18. Ebaid A, Rach R, El-Zahar E (2017) A new analytical solution of the hyperbolic Kepler equation using the Adomian decomposition method. Acta Astron 138:1–9

    Article  Google Scholar 

  19. Wazwaz AM (1999) A reliable modification of Adomian decomposition method. Appl Math Comput 102:77–86

    MathSciNet  MATH  Google Scholar 

  20. Ray SS, Bera RK (2005) An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl Math Comput 167:561–571

    MathSciNet  MATH  Google Scholar 

  21. Luo XG (2005) A two-step Adomian decomposition method. Appl Math Comput 170:570–583

    MathSciNet  MATH  Google Scholar 

  22. Wazwaz AM, El-Sayed SM (2001) A new modification of the Adomian decomposition method for linear and nonlinear operators. Appl Math Comput 122:393–405

    MathSciNet  MATH  Google Scholar 

  23. Assari P, Cuomo S (2019) The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines. Eng Comput 35:1391–1408

    Article  Google Scholar 

  24. Assari P, Mehregan FA (2019) Local radial basis function scheme for solving a class of fractional integro-differential equations based on the use of mixed integral equations. Z Angew Math Mech. https://doi.org/10.1002/zamm.201800236

    Article  MATH  Google Scholar 

  25. Assari P, Dehghan M (2019) A meshless local Galerkin method for solving Volterra integral equations deduced from nonlinear fractional differential equations using the moving least squares technique. Appl Num Math 143:276–299

    Article  MathSciNet  Google Scholar 

  26. Odibat Z, Momani S (2008) Numerical methods for nonlinear partial differential equations of fractional order. Appl Math ModeApp Math Mode 32:28–39

    Article  Google Scholar 

  27. Ding H, Li C (2016) A high-order algorithm for Riesz derivative and their applications. Frac Cal Appl Anal 19:19–55

    Article  MathSciNet  Google Scholar 

  28. Yu Y, Deng W, Wu Y (2014) Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations. Commun Math Sci 15:1183–1209

    Article  Google Scholar 

  29. Kumar K, Pandey RK, Sharma S (2017) Comparative study of three numerical schemes for fractional integro-differential equations. J Comput Appl Math 315:287–302

    Article  MathSciNet  Google Scholar 

  30. Palais RS (2007) A simple proof of the Banach contraction principle. J Fix Point Appl 2:221–223

    Article  MathSciNet  Google Scholar 

  31. Falset JG, Latrach K, Gàlvez EM, Taoudi MA (2012) Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness. J Differ Equ 252:3436–3452

    Article  MathSciNet  Google Scholar 

  32. Green JW, Valentine FA (2019) On the Arzelà-Ascoli Theorem. Math Mag 34:199–202

    MATH  Google Scholar 

  33. Ding H (2019) A high-order numerical algorithm for two-dimensional time-space tempered fractional diffusion-wave equation. Appl Num Math 135:30–46

    Article  MathSciNet  Google Scholar 

  34. Murad SA, Hadid SB (2012) Existence and uniqueness theorem for fractional differential equation with integral boundary condition. J Frac Cal Appl 3:1–9

    MATH  Google Scholar 

  35. Abdo MS, Saeed AM, Panchal SK (2019) Caputo fractional intergo-differential equation with non-local condition in Banach space. Int J Appl Math 32:279–288

    Article  Google Scholar 

  36. Dehghan M, Abbaszadeh M (2018) A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation. Comput Math Appl 75:2903–2914

    Article  MathSciNet  Google Scholar 

  37. Ray SS (2009) Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method. Commun Nonlinear Sci Num Simul 14:1295–1306

    Article  MathSciNet  Google Scholar 

  38. Chen M, Deng W (2017) A second-order accurate numerical method for the space-time tempered fractional diffusion-wave equation. Appl Math Lett 68:87–93

    Article  MathSciNet  Google Scholar 

  39. Oruc Ö, Esen A, Bulut F (2019) A Haar wavelet approximation for two-dimensional time fractional reaction-subdiffusion equation. Eng Comput 35:75–86

    Article  Google Scholar 

  40. Rach R (2012) A bibliography of the theory and applications of the Adomian decomposition method. Kybernetes 41:1961–2011

    Article  MathSciNet  Google Scholar 

  41. Hafez RM, Zaky MA (2019) High-order continuous Galerkin methods for multi-dimensional advection-reaction-diffusion problems. Eng Comput. https://doi.org/10.1007/s00366-019-00797-y

    Article  Google Scholar 

  42. Karamali G, Dehghan M, Abbaszadeh M (2019) Numerical solution of a time-fractional PDE in the electroanalytical chemistry by a local meshless method. Eng Comput 35:87–100

    Article  Google Scholar 

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Acknowledgements

We express our sincere thanks to the editor in chief, editor, and reviewers for their valuable suggestions to revise this manuscript.

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  1. Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj, Uttar Pradesh, 211004, India

    Pratibha Verma & Manoj Kumar

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  1. Pratibha Verma
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  2. Manoj Kumar
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Correspondence to Manoj Kumar.

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Verma, P., Kumar, M. Exact solution with existence and uniqueness conditions for multi-dimensional time-space tempered fractional diffusion-wave equation. Engineering with Computers 38, 271–281 (2022). https://doi.org/10.1007/s00366-020-01029-4

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