This paper presents a family of one-parameter fixed-point iterative schemes based on linear combinations, extending the classical framework of convex combination methods. Building upon this formulation, we propose a three-step iterative scheme that provides a structural generalization of several existing methods in the literature. Under appropriate contractive-type assumptions on Banach spaces, we establish theoretical results, including the strong convergence, stability, and data dependence of the proposed scheme. Numerical experiments on a variety of reference problems illustrate that the proposed method performs competitively with well-known iterative schemes.

This paper introduces and analyzes some new highly efficient iteration procedures for approximating fixed points of contractive-type mappings. The stability, strong convergence, and performance of the proposed schemes have been investigated through numerical examples. Numerical examples demonstrate that the newly introduced schemes produce highly accurate approximations comparable to other similar robust schemes appeared in the literature. Nevertheless, all the schemes developed here are more efficient than similar robust schemes available in the literature.

In this article, high temporal and spatial resolution schemes are combined to solve the Camassa-Holm and Degasperis-Procesi equations. The differential quadrature method is strengthened by using modified uniform algebraic trigonometric tension B-splines of order four to transform the partial differential equation (PDE) into a system of ordinary differential equations. Later, this system is solved considering an optimized hybrid block method. The good performance of the proposed strategy is shown through some numerical examples. The stability analysis of the presented method is discussed. This strategy produces a saving of CPU-time as it involves a reduced number of grid points.

In this study, the authors present a uniform algebraic trigonometric tension B‐spline‐based differential quadrature method combined with an optimized hybrid block method to numerically solve the Rosenau–KdV–RLW equation. The discrete mass and energy have been calculated, showing that they are conserved, thus indicating the efficiency and accuracy of the present approach. The method outperforms other methods and does not require linearization, allowing it to be directly implemented for solving nonlinear partial differential equations.

This paper introduces and analyzes some new highly efficient iterative procedures for approximating fixed points of contractive-type mappings. The stability, data dependence, strong convergence, and performance of the proposed schemes are addressed. Numerical examples demonstrate that the newly introduced schemes produce approximations of great accuracy and comparable to other similar robust schemes appeared in the literature. Nevertheless, all the schemes developed here are more efficient than other robust schemes used for comparison.

We propose and study a new Jungck-type iteration scheme to approximate coincidence points of contractive mappings. The strong convergence, stability, and data dependency results have been discussed. Numerical experiments demonstrate that the newly introduced Jungck-type iteration scheme yields a higher convergence rate in comparison with other Jungck-type iteration schemes available in the literature.

Investigation of the solutions of the coupled viscous Burgers system is crucial for realizing and understanding some physical phenomena in applied sciences. Particularly, Burgers equations are used in the modeling of fluid mechanics and nonlinear acoustics. In the present study, a modified meshless quadrature method based on radial basis functions is used to discretize the partial derivatives in the spatial variable. A technique to find the best value of the shape parameter is introduced. A high-resolution optimized hybrid block method is then used to solve the problem in the temporal variable. To validate the proposed method, several test problems are considered and the simulated results are compared with exact solutions and previous works. Moreover, a sensitivity analysis for parameter c is conducted, and the unconditional stability of the proposed algorithm has been validated.

This article introduces a computational hybrid one-step technique designed for solving initial value differential systems of a first order, which utilizes second derivative function evaluations. The method incorporates three intra-step symmetric points that are calculated to provide an optimum version of the suggested scheme. By combining the hybrid and block methodologies, an efficient numerical method is achieved. The hybrid nature of the algorithm determines that the first Dahlquist barrier is overcome, ensuring its effectiveness. The proposed technique exhibits an eighth order of convergence and demonstrates A-stability characteristics, making it particularly well suited for handling stiff problems. Additionally, an adjustable step size variant of the algorithm is developed using an embedded-type technique. Through numerical experiments, it is shown that the suggested approach outperforms some other well-known methods with similar properties when applied to initial-value ordinary differential problems.

This work proposes a new three-step iteration scheme to approximate the fixed points of a contractive like mapping. Further, the stability and strong convergence of the proposed scheme are considered, and their applicability to different problems is discussed. Some numerical experiments are presented, showing that the new scheme outperforms all the well-known existing three-step schemes available in the literature.

This article addresses the development and analysis of an efficient optimized hybrid block method for integrating general second order initial value problems (IVPs) of ordinary differential equations (ODEs).The construction of the method is based on a combination of two methodologies, namely hybrid and block that result in an efficient implicit numerical integrator.Further, an improved strategy is obtained considering its adaptive step-size formulation.Some numerical experiments have been carried out on solving some well-known problems existing in the literature with fixed and adaptive step-size implementation of the new scheme.Numerical data reveals that the new scheme is a good alternative to existing solvers with comparable properties.

In this paper, a new one-parameter class of fixed point iterative method is proposed to approximate the fixed points of contractive type mappings. The presence of an arbitrary parameter in the proposed family increases its interval of convergence. Further, we also propose new two-step and three-step fixed point iterative schemes. We also discuss the stability, strong convergence and fastness of the proposed methods. Furthermore, numerical experiments are performed to check the applicability of the new methods, and these have been compared with well-known similar existing methods in the literature.

In this paper, we present a new third-order family of iterative methods in order to compute the multiple roots of nonlinear equations when the multiplicity (m≥1) is known in advance. There is a plethora of third-order point-to-point methods, available in the literature; but our methods are based on geometric derivation and converge to the required zero even though derivative becomes zero or close to zero in vicinity of the required zero. We use the exponential fitted curve and tangency conditions for the development of our schemes. Well-known Chebyshev, Halley, super-Halley and Chebyshev–Halley are the special members of our schemes for m=1. Complex dynamics techniques allows us to see the relation between the element of the family of iterative schemes and the wideness of the basins of attraction of the simple and multiple roots, on quadratic polynomials. Several applied problems are considered in order to demonstrate the performance of our methods and for comparison with the existing ones. Based on the numerical outcomes, we deduce that our methods illustrate better performance over the earlier methods even though in the case of multiple roots of high multiplicity.

A novel combination of two schemes has been implemented to determine the approximate solution of Kuramoto–Sivashinsky equation. Differential quadrature method using well-known cubic trigonometric B-splines is adapted in space to obtain a system of initial value problems and reformulated one-step optimized hybrid block method is constructed to deal with the resulted system. A stability and convergence analysis of the method is discussed in detail. Numerical findings corroborate the accuracy and better performance of the proposed method.

Integrated pest management (IPM) strategy has greatly contributed to a progressive commitment for sustainable agriculture in India. Long term studies conducted on validation and promotion of IPM in basmati rice (Oryza sativa L.) in district Gautam Budh Nagar, Uttar Pradesh, India resulted in a gradual enhancement in the area under IPM from 40 ha in 2010 by participation of 25 farmers to 990 ha in 2019 by participation of 654 farmers from 42 villages. Implementation of IPM technology resulted in a significant (P<0.05) reduction in the incidence of yellow stem borer (69.64%), leaf folder (70.9%), brown plant hopper (55.52%), bakane (90.98%) and the population of root-knot nematode (76.8%) over farmers’ practices (FP). It enhanced the population of predatory spiders (84.2%), beneficial soilnematodes (159.27%) and bio-agents, viz. Pseudomonas fluorescens (78.74%) and Trichoderma harzianum (81.34%) over FP. Application of chemical pesticides was reduced to 75.25 g/ha in IPM against 892.93 g/ha in farmers’ practice. The maximum residue level of buprofezin, a widely used insecticide, was recorded below detectable level in paddy grains. Long term studies indicated higher yield (38.0 q/ha) as well as benefit:cost ratio (3.6) in IPM as compared to FP yield (30.5 q/ha) and benefit:cost ratio (2.3) with 58.3% enhancement in net return over FP. Thorough analysis of the data indicated the availability of critical inputs, accessibility of farmers to subject matter specialists through Farmer Field Schools and market for IPM produces as the main factors responsible for sustainability and horizontal spread of IPM in Gautam Budh Nagar.

In this article, a family of one-step hybrid block methods having two intrastep points is developed for solving first-order initial value stiff differential systems that occur frequently in science and engineering. In each method of the family, an intrastep point controls the order of the main method and a second one has a control over the stability features of the method. The approach used to develop the class of A -stable methods is based on interpolation and collocation procedures. The methods exhibit hybrid nature and produce numerical solutions at several points simultaneously. These methods can also be formulated as Runge-Kutta (RK) methods. Comparisons between the RK and block formulations of the proposed methods reveal a better performance of the block formulation in terms of computational efficiency. Furthermore, the efficiency of the methods is improved when they are formulated as adaptive step-size solvers using an error-control approach. Some methods of the proposed class have been tested to solve some well-known stiff differential systems. The numerical experiments show that the proposed family of methods performs well in comparison with some of the existing methods in the scientific literature.

In this paper, the numerical solution of a mixed derivative type Hunter–Saxton equation is addressed. A given equation is discretized transforming it into a system of ODEs with the use of a cubic trigonometric B-splines based differential quadrature method. The system is further solved using a fifth-order optimized one-step hybrid block method. Three numerical illustrations validate the efficiency of the proposed scheme and show its better performance through very accurate results. Stability and convergence analysis are also performed.

In this article, a two-parameter class of hybrid block methods for integrating first-order initial value ordinary differential systems is proposed. The methods exhibit hybrid nature which helps in bypassing the first Dahlquist barrier existing for linear multistep methods. The approach used in the development of a class of methods is purely interpolation and collocation technique. The class of methods is based on four intra-step points from which two intra-step points have been optimized by using an optimization strategy. In this optimization strategy, the values of two intra-step points are obtained by minimizing the local truncation errors of the formulas at the points xn+1/2 and xn+1.The order of accuracy of the proposed methods is six. A method as a special case of this class of methods is considered and developed into a block form which produces approximate numerical solutions at several points simultaneously. Further, the method is formulated into an adaptive step-size algorithm using an embedded type procedure. This method which is a special case of this class of methods has been tested on six well-known first-order differential systems.

This study presents a new one-parameter family of the well-known fixed point iteration method for solving nonlinear equations numerically. The proposed family is derived by implementing approximation through a straight line. The presence of an arbitrary parameter in the proposed family improves convergence characteristic of the simple fixed point iteration as it has a wider domain of convergence. Furthermore, we propose many two-step predictor–corrector iterative schemes for finding fixed points, which inherit the advantages of the proposed fixed point iterative schemes. Finally, several examples are given to further illustrate their efficiency.

In this article, we have considered an adaptive step-size formulation of an optimized block method for directly solving general second-order initial value problems of ODEs numerically. This formulation has been done using an embedded-type procedure resulting in an efficient method that performs much better compared to its counterpart fixed step-size method and other existing block strategies.

In this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.