In this paper, a Modified Variational Iterat ion Method (MVIM) for the solution of a differential equation of Bratu-type is presented. The method converges to the exact solution after an iteration. This shows that the method is efficient for this class of init ial and boundary value problems.

Background Experimentally brought to light by Russell and hypothetically explained by Korteweg–de Vries, the KDV equation has drawn the attention of several mathematicians and physicists because of its extreme substantial structure in describing nonlinear evolution equations governing the propagation of weakly dispersive and nonlinear waves. Due to the prevalent nature and application of solitary waves in nonlinear dynamics, we discuss the soliton solution and application of the fractional-order Korteweg–de Vries (KDV) equation using a new analytical approach named the “Modified initial guess homotopy perturbation.” Results We established the proposed technique by coupling a power series function of arbitrary order with the renown homotopy perturbation method. The convergence of the method is proved using the Banach fixed point theorem. The methodology was demonstrated with a generalized KDV equation, and we applied it to solve linear and nonlinear fractional-order Korteweg–de Vries equations, which are in Caputo sense. The method’s applicability and effectiveness were established as a feasible series of arbitrary orders that accelerate quickly to the exact solution at an integer order and are obtained as solutions. Numerical simulations were conducted to investigate the effect of Caputo fractional-order derivatives in the dispersion and propagation of water waves by varying the order $$\alpha$$ α on the $$[0,1]$$ [ 0 , 1 ] interval. Comparative analysis of the simulation results, which were presented graphically and discussed, reveals that the degree of freedom of the Caputo fractional-order derivative is vital to controlling the magnitude of environmental hazards associated with water waves when adjusted. Conclusion The proposed method is recommended for obtaining convergent series solutions to fractional-order partial differential equations. We suggested that applied mathematicians and physicists investigate this work to better understand the impact of the degree of freedom posed by Caputo fractional-order derivatives in wave dispersion and propagation, as physical applications can help divert wave-related environmental hazards.

In this paper, an improved algorithm for the solution of Generalized Burger-Fisher's Equation is presented. A Maple code is generated for the algorithm and simulated. It was observed that the algorithm gives the solution with less computation. The solution gives a better result when compared with the numerical solutions in the existing literature.

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Background The world's survival ability has been threatened by the COVID-19 outbreak. The possibility of the virus reemerging in the future should not be disregarded, even if it has been confined to certain areas of the world after wreaking such havoc. This is because it is impossible to prove that the virus has been totally eliminated. This research attempts to investigate the spread and control of the COVID-19 virus in Nigeria using the Caputo fractional order derivative in a proposed model. Results We proposed a competent nine-compartment model of Corona virus infection. It starts by demonstrating that the model is epidemiologically sound in terms of solution existence and uniqueness. The basic reproduction threshold R0 was determined using the next-generation matrix technique. We applied the Laplace-Adomian decomposition method to the fractional-order Caputo's derivative model of the Corona virus disease to produce the approximate solution of the model analytically. The obtained results, in the form of an infinite series, were simulated using the MAPLE 18 package to investigate the effect of fractional order derivative on the dynamics of COVID-19 transmission in the model and shed light on methods of eradication. The graphical interpretations of the simulation process were shown and discussed accordingly. Conclusions The study reveals the effect of the Caputo fractional order derivative in the transmission dynamics of the disease. Individual recovery was found to be greatest at an integer order, which represents the full implementation of other factors such as treatment, vaccination, and disease transmission reduction. Hence, we advised that researchers, government officials, and health care workers make use of the findings of this study to provide ways in which disease transmission will be reduced to a minimum to stop the prevalence of COVID-19 by applying the findings of this study.