A Reliable Method for Obtaining Analytical Solutions to Some Ordinary and Partial Differential Equations with Initial Values
Galal Mahrous Moatimid,
Mohamed Abdel-Latif Ramadan,
Mahmoud Hamed Taha,
Elsayed Eladdad
Issue:
Volume 7, Issue 2, June 2022
Pages:
20-30
Received:
19 February 2022
Accepted:
28 March 2022
Published:
20 April 2022
DOI:
10.11648/j.ajmcm.20220702.11
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Abstract: In this communication, the homotopy perturbation method is modified and extended to obtain the analytical solutions of some nonlinear differential equations. Differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc. Fluid mechanics, heat and mass transfer, and electromagnetic theory are all modeled by partial differential equations (PDE) and all have plenty of real life applications. dynamic meteorology and numerical weather forecasting: the weather report you see every night on TV has been obtained from the numerical solution of a complex set of nonlinear PDEs. The numerical solution of nonlinear differential equations is extremely difficult. Here, the proposed technique is implemented to obtain the analytical solutions of the initial-value ordinary and partial differential equations. In the current study, some problems are solved using a newly modified method that outperforms all other known methods, with approximate results in the form of power series. The method's algorithm is described and illustrated using some well-known problems. The obtained results demonstrate the method's efficiency. Furthermore, those results implies that this new method is simpler to implement. The approach is powerful, effective, and promising in analyzing some classes of differential equations for heat conduction problems and other dynamical systems. To crystallize the new approach, some illustrated examples are introduced.
Abstract: In this communication, the homotopy perturbation method is modified and extended to obtain the analytical solutions of some nonlinear differential equations. Differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat...
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Caputo Sense Fractional Order Derivative Model of Cholera
Sani Fakai Abubakar,
Mohammed Olarenwaju Ibrahim
Issue:
Volume 7, Issue 2, June 2022
Pages:
31-36
Received:
30 June 2022
Accepted:
26 July 2022
Published:
27 September 2022
DOI:
10.11648/j.ajmcm.20220702.12
Downloads:
Views:
Abstract: A deterministic mathematical cholera model is formulated using ordinary differential equations. The formulated system of equations was then transformed into fractional derivative of Caputo sense, with order λ that ranges between 0 and 1. The transformed equations were displayed in Caputo sense fractional order derivative using the fractional derivative operator. These equations were then interpreted and the numerical Adams-Bashforth-Moulton kind of predictor-corrector method was used on maple 18 software to obtain the model’s outcome. Dynamics of cholera disease controls, comprising treatment, hygiene consciousness and vaccine were analyzed and the results were produced in graphs. The graphs show the dynamics of the susceptible, effects of vaccine on the susceptible and the rate of cholera infection. After studying and interpretation of the graphs, the result show that lower fractional order values in the range 0.25 to 0.5 gives lower values of susceptible and vaccinated individuals but gives higher number of infected individuals. To test efficiency of the obtained result, we compared it with the integer order derivative result, and found that the fractional order results gave a better and efficient, portray of the successful useable controls. Caputo sense fractional order derivative using Adams-Bashforth-Moulton kind of predictor-corrector numerical method, guaranteed getting result similar to Runge-Kutta fourth-order numerical method.
Abstract: A deterministic mathematical cholera model is formulated using ordinary differential equations. The formulated system of equations was then transformed into fractional derivative of Caputo sense, with order λ that ranges between 0 and 1. The transformed equations were displayed in Caputo sense fractional order derivative using the fractional deriva...
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