Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method

Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay

“…In this paper, we obtain the numerical solution of initial valued autonomous system of TFPDEs with proportional delay [34] defined by…”

“…Among others, HPTM has been employed for solving fractional model of Navier-Stokes equation [25], optimal control problems [26], fractional coupled sineGordon equations [27], Falkner-Skan wedge flow [28], time-and space-fractional coupled Burgers' equations [29], strongly nonlinear oscillators [30], nonlinear boundary values problems [31,32], non-homogeneous partial differential equations with a variable coefficient [33]. The reader also refer to read [34].…”

“…To best of my knowledge, a little literature of numerical techniques to solve TFPDE with delay, among them, Chebyshev pseudospectral method for linear differential and differentialfunctional parabolic equations by Zubik-Kowal [37], spectral collocation & waveform relaxation methods by Zubik-Kowal and Jackiewicz [38] and iterated pseudospectral method [39] for nonlinear delay partial differential equations, two dimensional differential transform method (2D-DTM) and RDTM for partial differential equations with proportional delay by Abazari and Ganji [40], Abazari and Kilicman [41] used DTM for nonlinear integro-differential equations with proportional delay, group analysis method for nonhomogeneous mucilaginous Burgers equation with proportional delay due to Tanthanuch [42], homotopy perturbation method for TFPDE with proportional delay by Sakar et al [34] and Shakeri-Dehghan [43], and Biazar ad Ghanbari [44], variational iteration method (VIM) for solving a neutral functional-differential equation with proportional delays by Chena and Wang [45], functional constraints method for the exact solutions of nonlinear delay reaction-diffusion equations by Polyanin and Zhurov [46], and so on.…”

Improvement of the homotopy perturbation method to nonlinear problems

“…The HPM can be applied to integro-differential equation (Elbeleze et al, 2016), linear and nonlinear NewellWhitehead-Segel equations (Nourazar et al, 2017), nonlinear optimal control problems (Jafari et al, 2016), integral equations (Elzaki and Alamri, 2016;Hasan and Matin, 2017), nonlinear wave-like equations with variable coefficients (Gupta et al, 2013), boundary value problems (Opanuga et al, 2017), the quadratic Riccati differential equation (Aminikhah and Hemmatnezhad, 2010), Boussinesq-like equations (Fernández, 2014) and others (Sakar et al, 2016;Soori et al, 2015;Qureshi et al, 2017;Zhang et al, 2014;Najafi and Edalatpanah, 2014;Roy et al, 2015;Abou-Zeid, 2016).…”

Numerical solution of neutral functional-differential equations with proportional delays

“…Biazar and Ghanbari applied the homotopy perturbation method to obtain numerical solution of NFDEs with proportional delays [9] and so on [10,11].…”