In this article, a hybrid technique of Elzaki transformation and decomposition method is used to solve the Navier–Stokes equations with a Caputo fractional derivative. The numerical simulations and examples are presented to show the validity of the suggested method. The solutions are determined for the problems of both fractional and integer orders by a simple and straightforward procedure. The obtained results are shown and explained through graphs and tables. It is observed that the derived results are very close to the actual solutions of the problems. The fractional solutions are of special interest and have a strong relation with the solution at the integer order of the problems. The numerical examples in this paper are nonlinear and thus handle its solutions in a sophisticated manner. It is believed that this work will make it easy to study the nonlinear dynamics, arising in different areas of research and innovation. Therefore, the current method can be extended for the solution of other higher-order nonlinear problems.

Combining phosphorus (P) with phosphate solubilizing bacteria (PSB) improved wheat yield and P uptake in alkaline soil. AbstractPhosphate solubilizing bacteria can reduce dependence on chemical phosphorus (P) fertilizers by mineralizing and solubilizing indigenous soil P. What's why, we assessed the interactive effect of phosphate solubilizing bacteria (with and without PSB) and phosphorus levels (60, 90 and 120 kg P2O5 ha -1 ) on P uptake and yield of wheat crop under field conditions. Two factorial randomized complete block design (RCBD) with three replications was used. The PSB inoculation significantly enhanced plant height (3%), 1000 grains weight (12%), grain (6%), biological (13%) and straw yield (18.5%) of wheat over control. Inoculation with PSB also significantly improved plant P concentration and uptake (26% each) over un-inoculated control. Similarly, with increasing application rates of P from 60 to 120 kg P2O5 ha -1 the tested parameter were significantly improved except straw yield. The interactive effect of PSB and P exhibited significant effect on 1000 grains weight while the rest of parameter didn't respond significantly. However, generally PSB inoculation with P enhanced yield attributes and improved P use efficiency over sole application of P. Thus it may be concluded that PSB should be applied with P to enhance wheat yield and P use efficiency.

This comparative study of fractional nonlinear fractional Burger’s equations and their systems has been done using two efficient analytical techniques. The generalized schemes of the proposed techniques for the suggested problems are obtained in a very sophisticated manner. The numerical examples of Burger’s equations and their systems have been solved using Laplace residual power series method and Elzaki transform decomposition method. The obtained results are compared through graphs and tables. The error tables have been constructed to show the associated accuracy of each method. The procedures of both techniques are simple and attractive and, therefore, can be extended to solve other important fractional order problems.

In the present note, a new modification of the Adomian decomposition method is developed for the solution of fractional-order diffusion-wave equations with initial and boundary value Problems. The derivatives are described in the Caputo sense. The generalized formulation of the present technique is discussed to provide an easy way of understanding. In this context, some numerical examples of fractional-order diffusion-wave equations are solved by the suggested technique. It is investigated that the solution of fractional-order diffusion-wave equations can easily be handled by using the present technique. Moreover, a graphical representation was made for the solution of three illustrative examples. The solution-graphs are presented for integer and fractional order problems. It was found that the derived and exact results are in good agreement of integer-order problems. The convergence of fractional-order solution is the focus point of the present research work. The discussed technique is considered to be the best tool for the solution of fractional-order initial-boundary value problems in science and engineering.

In this article, a new modification of the Adomian decomposition method is performed for the solution fractional order convection–diffusion equation with variable coefficient and initial–boundary conditions. The solutions of the suggested problems are calculated for both fractional and integer orders of the problems. The series of solutions of the problems with variable coefficients have been provided for the first time. To verify and illustrate our new technique, four numerical examples are presented and solved by using the proposed technique. The derived results are plotted, and the dynamics are shown for both fractional and integer orders of the problems. An excellent variation among the solutions at various fractional orders is observed. It is analyzed that the new technique based on the Adomian decomposition method is accurate and effective. The present method fits both the initial and boundary conditions with double approximations simultaneously, which increases the accuracy of the present method. For the first time, the present technique is used for the solutions of the problems with variable coefficients along with initial and boundary conditions. It is therefore suggested to apply the present procedure for the solutions of other problems with variable order and coefficients along with initial and boundary conditions.