Muhammad Abbas

University of Sargodha | UOS · Department of Mathematics

This paper employs the generalized projective Riccati equation method and the Sardar sub-equation technique to extract the solitary wave solutions of the nonlinear (2+1)-dimensional Coupled Riemann wave equations, which describes the electrostatic and magneto-sound waves in plasma, ion cyclotron waves, tidal and tsunami waves, homogeneous and stati...

The present study employs the improved F-expansion and modified \(\exp (-Z(\varsigma ))\)-expansion function methodologies to generate an enormous amount of novel wave solutions for the \((2 + 1)\)-dimensional extended shallow water wave equation. This equation has widespread applications in many scientific and engineering domains, such as oceanogr...

This paper applies the extended direct algebraic method (EDAM) for the first time to obtain the soliton solutions of the fractional Sasa–Satsuma equation (FSSE) with the descriptions of two fractional derivatives. To evaluate the optical soliton and some other solutions of the FSSE, the presented research applies the fractional beta and conformable...

Nonlinear distinct models have wide applications in various fields of science and engineering. The present research uses the mapping and generalized Riccati equation mapping methods to address the exact solutions for the nonlinear Klein–Gordon equation. First, the travelling wave transform is used to create an ordinary differential equation form fo...

Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-integer-order derivatives. In the current research, an effective cubic B-spline collocation method is us...

The main aim of this study is to obtain soliton solutions of the generalized reaction Duffing model, which is a generalization for a collection of prominent models describing various key phenomena in science and engineering. The equation models the motion of a damped oscillator with a more complex potential than in basic harmonic motion. Two effect...

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trig...

In this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering,...

Citation: Vivas-Cortez, M.; Baloch, S.A.; Abbas, M.; Alosaimi, M.; Wei, G. Lump, Breather, Ma-Breather, Kuznetsov-Ma-Breather, Periodic Cross-Kink and Multi-Waves Soliton Solutions for Benney-Luke Equation. Symmetry 2024, 16, 747. https:// Abstract: The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soli...

Nonlinear partial differential equations (NLPDEs) are used in a wide range of natural and applied sciences phenomena. A fascinating and rapidly developing scientific field is the study of soliton solutions to nonlinear evolution equations (NLEEs). In this research, different types of soliton solutions for the strain wave model (SWM) are derived by...

In this article, Elzaki decomposition method (EDM) has been applied to approximate the analytical solution of the time-fractional gas-dynamics equation. The time-fractional derivative is used in the Caputo-Fabrizio sense. The proposed method is implemented on homogenous and non-homogenous cases of the time-fractional gas-dynamics equation. A compar...

In this article, the modified extended Fan sub-equation approach as an analytical tool to investigate the optical soliton solutions of the paraxial wave dynamical model with Kerr media. Indistinguishable and non-diffractive spatially scattered waves transmitting in the optical Kerr medium are explained by optical solitary waves. The solitons, singl...

The time fractional Schrödinger equation contributes to our understanding of complex quantum systems, anomalous diffusion processes, and the application of fractional calculus in physics and cubic B-spline is a versatile tool in numerical analysis and computer graphics. This paper introduces a numerical method for solving the time fractional Schröd...

The dynamic behavior of the Vakhnenko-Parkes equation is examined in this manuscript. This is an important subject because of its implications for comprehending intricate mathematical models describing traveling wave phenomena and solitons. The construction of traveling wave solutions for the Vakhnenko-Parkes equation in closed form is the main iss...

In this article, the Bernoulli sub-ODE and generalized Kudryashov methods have been successfully used to look for travelling wave solutions for the coupled non-linear evolution equations including the coupled Higgs equation and the Maccari system. The aforementioned approaches provide more new broad solutions than the previous ones now in use and a...

A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The...

Fractional order nonlinear partial differential equations are useful for detailing many practical models in multidisciplinary fields. In theory and numerical studies, recognizing solitary wave solutions to these equations is therefore preferable. The nonlinear properties of the fractional Peyrard Bishop DNA (FPBDNA) model make it relevant to severa...

The present research investigates the double-chain deoxyribonucleic acid model, which is important for the transfer and retention of genetic material in biological domains. This model is composed of two lengthy uniformly elastic filaments, that stand in for a pair of polynucleotide chains of the deoxyribonucleic acid molecule joined by hydrogen bon...

In this work, the (2+1)-dimensional Painlevé integrable Burgers equation is investigated. By applying a certain unified method, some analytical solutions, involving rational functions, trigonometric functions and hyperbolic functions, are achieved. In order to predict the wave dynamics, several three-dimensional and two-dimensional graphs and conto...

The primary objective of this research is to develop a mathematical model, analyze the dynamic occurrence of thermal shock and exploration of how thermal memory with moving line impact of heat transfer within biological tissues. An extended version of the Pennes equation as its foundational framework, a new fractional modelling approach called the...

A B-spline is defined by the degree and quantity of knots, and it is observed to provide a higher level of flexibility in curve and surface layout. The extended cubic B-spline (ExCBS) functions with new approximation for second derivative and finite difference technique are incorporated in this study to solve the time-fractional Allen–Cahn equation...

Curvature lines are special and important curves on surfaces. It is of great significance to construct developable surface interpolated on curvature lines in engineering applications. In this paper, the shape optimization of generalized cubic ball developable surface interpolated on the curvature line is studied by using the improved reptile search...

The aim of the present study is to identify multiple soliton solutions to the nonlinear coupled Broer-Kaup-Kupershmidt (BKK) system, including beta, conformable, local-fractional, and M-truncated derivatives. The coupled Broer-Kaup-Kupershmidt system is employed for modelling nonlinear wave evolution in mathematical models of fluid dynamics, plasmi...

The second order Burger’s equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative i...

The generalization of the classical FitzHugh–Nagumo model provides a more accurate description of the physical phenomena of neurons by incorporating both nonlinearity and fractional derivatives. In this article, we present a numerical method for solving the time-fractional FitzHugh–Nagumo equation (TFFNE) in the sense of the Atangana–Baleanu fracti...

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractio...

The dynamical behaviour of the (4+1)-dimensional fractional Fokas equation is investigated in this paper. The modified auxiliary equation method and extended (G G 2)-expansion method, two reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions used the...

The dynamical behaviour of the (4+1)-dimensional fractional Fokas equation is investigated in this paper. The modified auxiliary equation method and extended (G′G2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setleng...

Splines are piecewise polynomials that are as smooth as they can be without forming a single polynomial. They are linked at specific points known as knots. Splines are useful for a variety of problems in numerical analysis and applied mathematics because they are simple to store and manipulate on a computer. These include, for example, numerical qu...

This work discusses a ternary 4-point approximation subdivision technique with two properties, namely, convexity and monotonicity preservation. The fundamental contribution of this research article is to extract the conditions that assure the suggested subdivision scheme’s convexity and monotonicity. The methodology for extracting these conditions...

The time-fractional nonlinear Drinfeld–Sokolov–Wilson system, which has significance in the study of traveling waves, shallow water waves, water dispersion, and fluid mechanics, is examined in the presented work. Analytic exact solutions of the system are produced using the modified auxiliary equation method. The fractional implications on the mode...

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second ord...

This paper presents an efficient enhanced snake optimizer termed BEESO for global optimization and engineering applications. As a newly mooted meta-heuristic algorithm, snake optimizer (SO) mathematically models the mating characteristics of snakes to find the optimal solution. SO has a simple structure and offers a delicate balance between exploit...

In this work, the modified auxiliary equation method (MAEM) and the Riccati–Bernoulli sub-ODE method (RBM) are used to investigate the soliton solutions of the fractional complex coupled maccari system (FCCMS). Nonlinear partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilizing a travelling wav...

The majority of scientific fields employ differential equations involving fractional-order derivatives to understanda wide range of physical events. Fractional derivatives and integrals have the ability to cop with complicatedproblems. Various types of fractional-order derivative have been dicovered, like -fractional and M-truncatedderivative. In t...

The approximate degree reduction of ball non-uniform rational b-splines (NURBS) curves is a pivotal and knotty technique in Computer-Aided Design (CAD)/Computer Aided Manufacture (CAM). As we all know, the multi-degree reduction of NURBS ones is a mathematical optimization problem that a swarm intelligence algorithm can deal with. This paper uses a...

The main objective of current paper is to examine the impacts of fractional parameters on the dynamic response of soliton waves in a nonlinear time-fractional thin-film ferroelectric materials equation (TFFEME). To achieve such a goal, the TFFEME is first rehabilitated into ordinary differential equation using a complex wave transformation. Solitar...

Most of the nonlinear phenomena are described by partial differential equation in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, biology and mathematical finance. The solutions of a wide range of nonlinear evolution equations exhibit the wave behavior corresponding to the underly...

The optimal multi-degree reduction of ball Said–Ball curves is an unsolved and knotty important technique in computer aided design (CAD) and computer graphics (CG) and is potentially used in many engineering fields involving geometric modeling. In this paper, an improved chimp optimization algorithm (ICHOA, for short) is used to solve the degree re...

Jellyfish search (JS) algorithm impersonates the foraging behavior of jellyfish in the ocean. It is a new developed meta-heuristic algorithm that solves complex and real world optimization problems. The global explore capability and robustness of JS are strong, but JS still has great development space in solving complex optimization problems with h...

The Matlab source code of the ICHOA related to the article "Hybrid chimp optimization algorithm for degree reduction of ball Said-Ball curves".

This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ6-model technique is one of t...

This manuscript aims to investigate the velocity profile for the blood flow through an artery subject to magnetic field. It has been investigated how periodic acceleration of the body and slip conditions affect the irregular pulsatile blood flow across a porous media inside an artery if a magnetic field is present, under the assumption that blood i...

This paper presents a study of the unsteady flow of second grade fluid through a capillary tube, caused by sinusoidally varying pressure gradient, with fractional derivative model. The fractional derivative is taken in Caputo-Fabrizio sense. The analytical solution for the velocity profile has been obtained for non-homogenous boundary conditions by...

The soliton solutions are one of the stable solutions where nonlinearity and dispersion are perfectly balanced. They are used in a wide variety of physical fields, including plasma, solid state, neuronal, biological production, and diffusion processes. Different analytical methods have been used until now to obtain the soliton solutions of the Sawa...

The purpose of this work is to study the memory effect analysis of Caputo–Fabrizio time fractional diffusion equation by means of cubic B-spline functions. The Caputo–Fabrizio interpretation of fractional derivative involves a non-singular kernel that permits to describe some class of material heterogeneities and the effect of memory more effective...

Special Issue Information Dear Colleagues, Nonlinearity occurs in all complex real-life phenomena. The construction and investigation of nonlinear mathematical models that arise in physics, bio-engineering, optics, fluid dynamics and other fields of science and engineering is necessary to understand the physical framework of the related real-life...

BEESO: Multi-strategy Boosted Snake-inspired Optimizer for Engineering Applications

In this paper, the Peyrard–Bishop–Dauxois model of DNA dynamics is discussed along with the fractional effects of the M-truncated derivative and β-derivative. The Kudryashov’s R method was applied to the model in order to obtain a solitary wave solution. The obtained solution is explained graphically and the fractional effects of the β and M-trunca...

The aim of this manuscript is to find the analytical solutions of the Korteweg–de Vries (KdV) equation with dual power law nonlinearity using the exp-function method. The KdV equation is used to study the shallow water wave behavior of long wavelengths and small amplitude. With the help of the exp-function method, a variety of exact wave solutions...

Numerous fields, including the physical sciences, social sciences, and earth sciences, benefit greatly from the application of fractional calculus (FC). The fractional-order derivative is developed from the integer-order derivative, and in recent years, real-world modeling has performed better using the fractional-order derivative. Due to the flexi...

In this study, the time-dependent potential coefficient in a higher-order PDE with initial and boundary conditions is numerically constructed for the first time from a nonlocal integral condition. Even though the inverse identification problem investigated in this study is ill-posed, it has a unique solution. For discretizing the direct problem and...

In this article, an analytical technique based on unified method is applied to investigate the exact solutions of non-linear homogeneous evolution partial differential equations. These partial differential equations are reduced to ordinary differential equations using different traveling wave transformations, and exact solutions in rational and pol...

The B-spline or the basis spline function, is a piecewise polynomial function made up of polynomials, its domain is subdivided by knots, and basis functions are non-zero on the entire domain. In this study, a new cubic B-spline (NCBS) approximation together with the θ-weighted scheme is formed to approximate the numerical solution of the time fract...

In the manufacturing process of free-form complex surfaces, various problems occur which degraded the performance and can be resolved by using an optimization technique. Thousands of real-world problems can be converted into optimization techniques with distinct objective functions to acquire the optimal solution. In this paper, an assembly of GHT-...

Cranial implants, especially custom made implants, are complex, important and necessary in craniofacial fracture restoration surgery. However, the classical procedure of the manual design of the implant is time consuming and complicated. Different computer-based techniques proposed by different researchers, including CAD/CAM, mirroring, reference s...

In this paper, we considered an inverse problem of recovering the time-dependent potential coefficient, for the first time, in the sixth-order Boussinesq-type equation from additional data as an over-specification condition. The unique solvability theorem for this inverse problem is supplied. However, since the governing equation is yet ill-posed (...

This paper introduces new trigonometric basis functions (TBF) in polynomial and rational form with two shape parameters (SPs). Some classical characteristics, such as the partition of unity, positivity, symmetry, CHP, local control and invariance under affine transformation properties are proven mathematically and graphically. In addition, differen...

Generalized Bernstein-like functions (gB-like functions) with different shape parameters are used in this work. Parametric and geometric conditions in generalized form are developed. Some numerical examples of the parametric continuity (PC) and geometric continuity (GC) constraints of generalized Bézier-like curves (gB-like curves) are analyzed wit...

Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline technique (SPST)....

This article aims to produce efficient numerical results using the Galerkin approach and quartic B-spline functions for the one-dimensional second-order nonlinear evolution equation. The quartic B-spline functions are used as the shape and weight functions for the Galerkin method. The time derivative and space parameters are discretized by the fini...

Because of its good geometric characteristics, Said-Ball curve has become a useful tool for shape design and geometric representation in product manufacturing. In this paper, an enhanced chimp optimization algorithm (CHOA, for short) is used to solve the problem of approximate multi-degree reduction of Said-Ball curve. Firstly, two strategies are u...

The current paper uses the cubic B-spline functions and θ-weighted scheme to achieve numerical solutions of the time fractional Burgers’ equation with Atangana-Baleanu derivative. A non-singular kernel is involved in the Atangana-Baleanu fractional derivative. For discretization along temporal and spatial grids, the proposed numerical technique emp...

Bézier curves and surfaces with shape parameters have received more attention in the field of engineering and technology in recent years because of their useful geometric properties as compared to classical Bézier curves, as well as traditional Bernstein basis functions. In this study, the generalized Bézier-like curves (gBC) are constructed based...