This work uses the modified Extended Direct Algebraic Method (mEDAM) with conformable derivatives to obtain accurate solutions for the diffusion-reaction equation with cubic nonlinearity and the nonlinear fractional generalised density-independent DR problem. We use fractional derivatives in the conformable sense to achieve precise polynomial-form solutions by converting the equations into autonomous two-dimensional plane systems. The technique solves nonlinear ordinary differential equations (ODEs) produced by a fractional transformation to produce travelling wave solutions. Through the visualisation of various solution profiles, as periodic, shock, kink, anti-kink, and soliton waves, numerical simulations offer important insights into the dynamics of nonlinear fractional partial differential equations. This work builds on earlier research and broadens our understanding of complex events.

In this paper, we introduce a new three-step Newton method for solving a system of nonlinear equations. This new method based on Gauss quadrature rule has sixth order of convergence (with $n=3$). The proposed method solves nonlinear boundary-value problems and integral equations in few iterations with good accuracy. Numerical comparison shows that the new method is remarkably effective for solving systems of nonlinear equations.

This study generates and investigates spreading solitons in the fractional DR quadratic equation (FDE) with fractional derivatives using the Extended Direct Algebraic Method (EDAM). In population growth, mathematical biology, and reaction-diffusion mechanisms, the FDE is crucial. Applying the series form solution to the NODE from the FDE conversion into a recommended EDAM yields many traveling soliton solutions. To characterize and explore soliton structure propagation, we draw shock and kink soliton solutions. Through reaction-diffusion mechanics and mathematical biology, we may explain complex processes in many academic subjects.

In this paper, we suggest a Newton-type method for solving the system of absolute value equations. This new method is a two-step method with the generalized Newton method as predictor. Convergence of the proposed method is proved under some suitable conditions. At the end, we take several numerical examples to show that the new method is very effective.

Purpose: This study presents a two-step numerical iteration method specifically designed to solve absolute value equations. The proposed method is valuable and efficient for solving absolute value equations. Several numerical examples were taken to demonstrate the accuracy and efficiency of the proposed method.Design/methodology/approach: We present a two-step numerical iteration method for solving absolute value equations. Our two-step method consists of a predictor-corrector technique. The new method uses the generalized Newton method as the predictor step. The four-point open Newton-Cotes formula is considered the corrector step. The convergence of the proposed method is discussed in detail. This new method is highly effective for solving large systems due to its simplicity and effectiveness. We consider the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solve it using the proposed method.Findings: The paper provides empirical insights into how to solve a system of absolute value equations.Originality/value: This paper fulfills an identified need to study absolute value equations.