Chemical turbulence equivalent to Nikolaevskii turbulence

The Sumudu homotopy perturbation method (SHPM) is applied to solve fractional order nonlinear differential equations in this paper.The current technique incorporates two notable  strategies in particular Sumudu transform (ST) and homotopy perturbation method (HPM). The  proposed method’s hybrid property decreases the number of the quantity of computations and  materials needed. In this method, illustration examples evaluate the accuracy and applicability of the mentioned procedure. The outcomes got by FSHPM are in acceptable concurrence with the specific  arrangement of the problem.

https://ijnaa.semnan.ac.ir/article_5094_1c0804b82b17a21154efa934c92fab57.pdf

The approximate analytical solutions of nonlinear fractional ordinary differential equations

This paper shows how to use the fractional Sumudu homotopy perturbation technique (SHP) with the Caputo fractional operator (CF) to solve time fractional linear and nonlinear partial differential equations. The Sumudu transform (ST) and the homotopy perturbation technique (HP) are combined in this approach. In the Caputo definition, the fractional derivative is defined. In general, the method is straightforward to execute and yields good results. There are some examples offered to demonstrate the technique's validity and use.

Time-Fractional Differential Equations with an Approximate Solution

The Fokker–Planck (FP) equation is of significant relevance in stochastic dynamics. To solve the FP equation, this paper proposes a physically guided deep learning-based method. The proposed method uses physical modeling as the constraint for guiding the deep neural networks to learn the solutions of the FP equation from unlabeled data. Four numerical examples were used to verify the accuracy and effectiveness of the proposed method for solving linear FP equations and nonlinear FP equations. Moreover, we examined the robustness of the trained model. Finally, extra observation data was used to replace the boundary conditions and initial conditions in complicated forms. Meanwhile, the effect of missing boundary conditions or initial conditions to the proposed method was further analyzed. The results demonstrated that observation data can be used as promising substitution for the initial and boundary conditions while the proposed method can still obtain accurate numerical solutions in the absence of some conditions. • A physically guided deep learning solver is proposed for solving the time-dependent FP equations. • It can calculate the numerical solution for each point while obtaining the stability of the solver at that point. • It allows the use of experimental observations in place of missing initial and boundary conditions. 

Physically guided deep learning solver for time-dependent Fokker–Planck equation

Recently, researchers have been interested in studying fractional differential equations and their solutions due to the wide range of their applications in many scientific fields. In this paper, a new approach called the Hussein–Jassim (HJ) method is presented for solving nonlinear fractional ordinary differential equations. The new method is based on a power series of fractional order. The proposed approach is employed to obtain an approximate solution for the fractional differential equations. The results of this study show that the solutions obtained from solving the fractional differential equations are highly consistent with those obtained by exact solutions.

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A New Approach for Solving Nonlinear Fractional Ordinary Differential Equations

A Novel Method for the Analytical Solution of Partial Differential Equations Arising in Mathematical Physics

Range of validity and intermittent dynamics of the phase of oscillators with nonlinear self-excitation

In this paper, we investigate solutions of nonlinear fractional differential equations by using Natural homotopy perturbation method (NHPM). This method is coupled by the Natural transform (NT) and homotopy perturbation method (HPM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the presented method.

https://ijnaa.semnan.ac.ir/article_4936_fba1a9499f3844a5d3b42d9c8a3ac4c3.pdf

Natural homotopy perturbation method for solving nonlinear fractional gas dynamics equations

We analyse a model of the fluid flow between elastic walls simulating arteries actively interacting with the blood. The lubrication theory for the flow is coupled with the pressure and shear stress from the walls. The resulting nonlinear partial differential equation describes the displacement of the walls as a function of the distance along the flow and time. The equation is solved numerically using the one-dimensional integrated radial basis function network method. A solution in the form of a self-sustained train of pulses is obtained. Numerical experiments demonstrate the process of formation of the train from randomly chosen initial conditions. Dependence of the pulses on the boundary conditions is explored.

 References S. J. Sherwin, L. Formaggia, J. Peiro and V. Franke. Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Int. J. Numer. Meth. Fluids , 43:673–700, 2003. doi:10.1002/fld.543 C. Kleinstreuer. Biofluid Dynamics . Taylor and Francis, 2006. M. S. Olufsen, C. S. Peskin, W. Y. Kim, E. M. Pedersen, A. Nadim and J. Larsen. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Eng. , 28:1281–1299, 2000. doi:10.1114/1.1326031 K. S. Matthys, J. Alastruey, J. Peiro, A. W. Khir, P. Segers, P. R. Verdonck, K. H. Parker and S. J. Sherwin. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech. , 40:3476–3486, 2007. doi:10.1016/j.jbiomech.2007.05.027 A. J. Roberts. A One-dimensional Introduction to Continuum Mechanics . World Scientific. Singapore, 1994. doi:10.1142/2496 D. V. Strunin, Fluid flow between active elastic plates. ANZIAM J. , 50:C871–C883, 2009. doi:10.21914/anziamj.v50i0.1452 D. V. Strunin, Autosoliton model of the spinning fronts of reaction. IMA J. Appl. Math. , 63:163–177, 1999. doi:10.1093/imamat/63.2.163 R. Huang and Z. Suo. Wrinkling of a compressed elastic film on a viscous layer. J. Appl. Phys. , 91:1135–1142, 2002. doi:10.1063/1.1427407 J. King. The isolation oxidation of silicon: The reaction-controlled case. SIAM J. Appl. Math. 49:1064–1080, 1989. doi:10.1137/0149064 D. Ngo-Cong, N. Mai-Duy, W. Karunasena and T. Tran-Cong. Local moving least square-one-dimensional integrated radial basis function networks technique for incompressible viscous flows. Int. J. Numer. Meth. Fluids , 70:1443–1474, 2012. doi:10.1002/fld.3640 P. Le, N. Mai-Duy, T. Tran-Cong and G. Baker. A meshless modeling of dynamic strain localization in quasi-brittle materials using radial basis function networks, CMES–Comp. Model. Eng. , 25:43–66, 2008. doi:10.3970/cmes.2008.025.043 N. Mai-Duy and T. Tran-Cong. Numerical solution of differential equations using multiquadric radial basis function networks, Neural Networks , 14:185–199, 2001. doi:10.1016/S0893-6080(00)00095-2

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Mayada Gassab Mohammed

Papers

A Novel Method for the Analytical Solution of Partial Differential Equations Arising in Mathematical Physics

Range of validity and intermittent dynamics of the phase of oscillators with nonlinear self-excitation

Natural homotopy perturbation method for solving nonlinear fractional gas dynamics equations

Numerical solution for the fluid flow between active elastic walls

A Modification Fractional Homotopy Analysis Method for Solving Partial Differential Equations Arising in Mathematical Physics