A new higher-implicit block method for the direct numerical solution of fourth order ordinary differential equation is derived in this research paper. The formulation of the new formula which is 15-step, is achieved through interpolation and collocation techniques. The basic numerical properties of the method such as zero-stability, consistency and A-stability have been examined. Investigation showed that the new method is zero stable, consistent and A-stable, hence convergent. Test examples from recent literature have been used to confirm the accuracy of the new method.
Two numerical methods- I2BBDF2 and I22BBDF2 that compute two points simultaneously at every step of integration by first providing a starting value via fourth order Runge-Kutta method are derived using Taylor series expansion. The two-point block schemes are derived by modifying the existing I2BBDF (5) method of Mohamad et al., (2018). Convergence and stability analysis of the new methods are established with the methods being of order two and A-stable in both cases. Despite the very low order of the new methods, the accuracy of these methods on some stiff initial value problems in the literature proves their superiority over existing methods of higher orders such as I2BBDF(5), BBDF(5), E2OSB(4) among others.
In this research we formulated the Plants diseases model with the aim of studying the dynamics of the use of lysobacter antibioticus for prevention and control of rice bacterial blight. The disease free equilibrium state of the models was also obtained by equating each of the equation of the modified model to zero and simplifying. The basic reproduction number for the model was derived using the next generation matrix approach. Numerical simulation was carried out using MATLAB2018a to virtualize the dynamics of the model. Five numerical experiment was carried out and it was shown that biocontrol help to reduce the population of the pathogen as well as act as treatment for those that are already exposed or infected with the disease. It was also observed that the biocontrol agent provide immunity to rice plants against been infected with the disease. Finally, we observed from the simulation that the earlier the control is introduced the more protection plants will receive.
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