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In this article, we successfully obtain novel solutions for the coupled Drinfel'd–Sokolov–Wilson DSW system utilizing various methods. These include soliton solutions characterized by hyperbolic, rational, and trigonometric functions. Specifically, the generalized exponential rational function method (GERFM) and a modified version of the new Kudryashov method (MVNK) are employed to derive diverse soliton solutions for the system. Additionally, we demonstrate numerical solutions for the coupled Drinfel'd–Sokolov–Wilson system using adaptive moving mesh and uniform mesh methods. Also, we study the stability and error analysis of the numerical schemes. To validate the accuracy and reliability of the exact solutions obtained through analytical methods, we compare them with the numerical solutions both analytically and graphically. The techniques presented in this article are deemed suitable and acceptable and can be effectively applied to solve other nonlinear evolution systems. [ABSTRACT FROM AUTHOR]

In this work, we consider a system of Volterra integro-differential equations with initial conditions. The derivative term in equations is multiplied with distinct small positive parameters giving rise to overlapping layers. We propose a numerical scheme that avoids the extra condition on the problem's data required by the scheme of Liang et al. (Comput Appl Math 39:255, 2020). We derive a priori and a posteriori error bounds for the proposed scheme and further rectify the shortcomings of a posteriori error estimation in Liang et al. (Comput Appl Math 39:255, 2020). Numerical results are presented in the form of graphs and tables that validate the proposed theory. [ABSTRACT FROM AUTHOR]

In this work, we propose a /methodology for the selection and refinement of finite elements for the integration of process design and control. The proposed methodology is based on the selection criteria of the Hamiltonian function through the implementation of the Pontryagin's minimum principle. The Hamiltonian function features to be continuous and constant over time for autonomous systems; nevertheless, the Hamiltonian function shows a nonconstant profile for underestimated discretization meshes, which is exploited in this work for the refinement of the discretization. Furthermore, the residuals at noncollocation points are evaluated to estimate the collocation error, this is used as a second refinement criterion in the proposed framework. The methodology is illustrated using two case studies featuring a reaction system with two CSTRs in series and the Williams–Otto reactor, respectively. The results showed that an accurate selection of the finite elements return economically attractive designs with fewer elements than those obtained with equidistributed finite element strategies. [ABSTRACT FROM AUTHOR]

The leading objective of this article is to investigate the analytical and numerical solutions of the Generalized Benjamin-Bona-Mahony (GBBM) equation. We also aim to compare the performance of the considered methods for solving this equation. The exact solution is obtained analytically while the numerical solutions are demonstrated using some techniques, namely, the adaptive moving mesh and uniform mesh methods. The exact solution is presented in a form of convergent power series. The finite differences are also applied to discretise the BBM equation. Under a suitable selection of parameters, some 2D and 3D surfaces for the obtained theoretical and numerical results are shown to compare the exact and numerical solutions.

An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual‐type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method. [ABSTRACT FROM AUTHOR]

The Kortweg–de Vries (KdV) equation is more appropriate to simulate some natural phenomena and gives more accurate results for some physical systems such as the movement of water waves. In this work, novel analytical traveling wave solutions for a nonlinear KdV system are explored using the sech method. The exact solutions are presented in the form of hyperbolic functions. These solutions show the propagation of water waves on the surface. We also implement the numerical adaptive moving technique (MMPDEs) to construct the relevant system’s numerical solutions. A detailed comparison between the numerical and analytical solutions is also presented to confirm the accuracy of the numerical technique. We present some new 2D and 3D figures to illustrate the behaviour of the exact and numerical solutions. The obtained numerical solutions verify the accuracy of the considered methods qualitatively and quantitatively. The stability of the obtained solutions is investigated using the Hamiltonian system. The achieved results can be applied for some new observations in the ocean, coastal water, macroscopic phenomena, and processes. The proposed techniques are potent tools to solve many other non-linear partial differential equations in applied mathematics.

A nonlinear fractional differential equation with Caputo fractional derivative is considered. The problem is discretized by an upwind finite difference scheme for which a posteriori error analysis in the maximum norm is derived. A partly heuristic argument based on this a posteriori error estimation leads to several suitable monitor functions, and a new monitor function is constructed to design an adaptive grid algorithm. Numerical results are presented to illustrate the performance of the presented adaptive method. Compared with the other monitor functions, the presented adaptive grid method based on the new monitor function is more suitable to solve this type of nonlinearfractional differential equation. [ABSTRACT FROM AUTHOR]

The Kortweg–de Vries (KdV) equation is more appropriate to simulate some natural phenomena and gives more accurate results for some physical systems such as the movement of water waves. In this work, novel analytical traveling wave solutions for a nonlinear KdV system are explored using the sech method. The exact solutions are presented in the form of hyperbolic functions. These solutions show the propagation of water waves on the surface. We also implement the numerical adaptive moving technique (MMPDEs) to construct the relevant system's numerical solutions. A detailed comparison between the numerical and analytical solutions is also presented to confirm the accuracy of the numerical technique. We present some new 2D and 3D figures to illustrate the behaviour of the exact and numerical solutions. The obtained numerical solutions verify the accuracy of the considered methods qualitatively and quantitatively. The stability of the obtained solutions is investigated using the Hamiltonian system. The achieved results can be applied for some new observations in the ocean, coastal water, macroscopic phenomena, and processes. The proposed techniques are potent tools to solve many other non-linear partial differential equations in applied mathematics. [ABSTRACT FROM AUTHOR]

Abstract The principal objective of the present paper is to manifest the exact traveling wave and numerical solutions of the good Boussinesq (GB) equation by employing He’s semiinverse process and moving mesh approaches. We present the achieved exact results in the form of hyperbolic trigonometric functions. We test the stability of the exact results. We discretize the GB equation using the finite-difference method. We also investigate the accuracy and stability of the used numerical scheme. We sketch some 2D and 3D surfaces for some recorded results. We theoretically and graphically report numerical comparisons with exact traveling wave solutions. We measure the L 2 $L_{2}$ error to show the accuracy of the used numerical technique. We can conclude that the novel techniques deliver improved solution stability and accuracy. They are reliable and effective in extracting some new soliton solutions for some nonlinear partial differential equations (NLPDEs).

The principal objective of this paper is to employ the $\exp (-f(\zeta )) $-expansion and the adaptive moving mesh methods to express the exact travelling wave solutions and the numerical solutions, respectively, for the variant Boussinseq equations. Hyperbolic tangent and cotangent functions are utilized to build the exact solutions. The used numerical method uses the finite differences to discretize the proposed equations. The obtained numerical results are compared with other results obtained using the uniform mesh method. The achieved results show that both solutions match with each other. We also illustrate some 2D and 3D figures to confirm the validity of the numerical approach applied here.

The leading objective of this article is to investigate the analytical and numerical solutions of the Generalized Benjamin-Bona-Mahony (GBBM) equation. We also aim to compare the performance of the considered methods for solving this equation. The exact solution is obtained analytically while the numerical solutions are demonstrated using some techniques, namely, the adaptive moving mesh and uniform mesh methods. The exact solution is presented in a form of convergent power series. The finite differences are also applied to discretise the BBM equation. Under a suitable selection of parameters, some 2D and 3D surfaces for the obtained theoretical and numerical results are shown to compare the exact and numerical solutions. [ABSTRACT FROM AUTHOR]

This paper presents an adaptive mesh method for solving the Korteweg- de Vries-Burgers (KdVB) equation numerically. The r-adaptive mesh method which employs a monitor function and one of the mesh equations (MMPDEs) presented by [11] which uses to concentrate and move the mesh. This technique correctly resolves the various structures seen in these problems. Furthermore, it decreases the computational effort in comparison to the numerical solution using a fixed uniform mesh scheme. Here, the exact solution of the KdVB equation at a fixed time is utilized to compare with that employing an adaptive moving mesh and a uniform mesh. By comparing the relative error for the obtained solutions of the problem, I will check the accuracy of the adaptive method. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel numerical method that redistributes unevenly given points on an evolving closed curve to satisfy equi-arclength(-like) condition. Without substantial difficulty, it is also capable of remeshing or employing adaptive mesh refinement. The key idea is to find the discrete inverse of the arclength(-like) function in the framework of the Fourier spectral method to obtain overall spectral accuracy. Both equi-arclength and curvature-dependent redistributions are extensively studied, and their spectral accuracy is verified by application to smoothly perturbed points on various curves. We further confirm that our method converges even for the points being perturbed nonsmoothly and randomly. To leverage the robustness of our method, a remeshing technique is applied in which the accuracy is not affected. Application to a periodic planar curve without any modification of our algorithm is also discussed. Then, to show the practical applicability, an evolving curve with large deformation is studied by coupling with point redistribution and remeshing in various flows such as mean curvature flow, Willmore flow, and Stokes flow. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution to the regularized long wave (GRLW) equation is investigated numerically applying two different techniques a Parabolic Monge Ampere (PMA) moving mesh and uni- form mesh. The PMA moving mesh is a method for generating a moving mesh which moves as the solution moves with time. The gen- erated mesh is obtained by having the gradient of a grid potential function. I use here the centred finite differences for both schemes. The comparison between these schemes is shown in the last table and figure. The analytical solution is derived in Appendix A. I find a so- lution when it has only one solitary wave. Next, I show when it has two or three solitary waves. Figures 2D and 3D shows all of these nu- merical results using the mentioned schemes compared with the exact solution. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution of a nonlinear first-order singularly perturbed differential equation with integral boundary condition is considered. The discrete method is generated by a backward Euler formula and the grid is obtained by equidistributing a monitor function based on arc-length. We first give a rigorous error analysis for the numerical method of this problem on a grid that is constructed adaptively from a knowledge of the exact solution. A first-order rate of convergence, independent of the perturbation parameter, is established. Then, an a posteriori error bound and the corresponding convergence result are derived for the presented numerical scheme on an adaptive grid, which is constructed adaptively from a discrete approximation of the exact solution. At last, numerical experiments are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The principal objective of this paper is to employ the exp ⁡ (− f (ζ)) -expansion and the adaptive moving mesh methods to express the exact travelling wave solutions and the numerical solutions, respectively, for the variant Boussinseq equations. Hyperbolic tangent and cotangent functions are utilized to build the exact solutions. The used numerical method uses the finite differences to discretize the proposed equations. The obtained numerical results are compared with other results obtained using the uniform mesh method. The achieved results show that both solutions match with each other. We also illustrate some 2D and 3D figures to confirm the validity of the numerical approach applied here. [ABSTRACT FROM AUTHOR]

Abstract Motivated by the moving mesh method in conventional finite element analysis, in this paper, we propose an r -adaptive isogeometric analysis framework by Winslow's mapping and monitor function approach. We construct a new monitor function based on the absolute curvature metric of the isogeometric solution surface. In the proposed r -adaptive framework, the redistribution of the inner control points is performed to increase the density of control points in the regions where curvature based or error based monitor functions are maximum. The new locations of the inner control points are obtained by solving governing PDEs using isogeometric collocation method which is more efficient in solving large-scale problems. By employing Winslow's mapping we ensure that the generated r -adaptive parameterization is valid for isogeometric analysis and can be used as optimal initial grid for subsequent h -refinements. Multiple two-dimensional heat conduction examples are solved which confirm the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we establish exact solutions for the non-linear coupled KdV equations. The exp-function method is used to construct the solitary travelling wave solutions for these equations. The numerical adaptive moving mesh PDEs (MMPDEs) method is also implemented in order to solve the proposed coupled KdV equations. The achieved results may be applicable to some plasma environments, such as ionosphere plasma. Some numerical simulations compared with the exact solutions are provided to illustrate the validity of the proposed methods. Furthermore, the modulational instability is analyzed based on the standard linear-stability analysis. The depiction of the techniques are straight, powerful, robust and can be applied to other nonlinear systems of partial differential equations.