Dimensional Nonlinear Partial Differential Equations Research Articles

The exact solutions to the generalized (2+1)-dimensional nonlinear wave equation

Due to the importance of the nonlinear partial differential equations in applied physics and engineering, many mathematicians and physicists are interesting to the nonlinear partial differential equations. One of the main tasks of studying the nonlinear partial differential equations is to obtain the exact solutions for the nonlinear partial differential equations. In this paper, we study the exact solutions of a generalized (2+1)-dimensional nonlinear partial differential equation. According to the modified hyperbolic function method and the traveling transformation method, the generalized (2+1)-dimensional nonlinear partial differential equation is changed into an ordinary differential equation. By taking the homogeneous balance between the highest order nonlinear terms and the highest order derivative terms, we change the differential equation into an algebraic system. By solving the algebraic system, we obtain the solutions for the nonlinear partial differential equation. At last, some solutions and their figures are given by specific parameters. All the solutions conclude the trigonometric function solutions, the positive hyperbolic function solutions, and the hyperbolic trigonometric function solutions. These solutions exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

Periodic, n-soliton and variable separation solutions for an extended (3+1)-dimensional KP-Boussinesq equation

An extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation is studied in this paper to construct periodic solution, n-soliton solution and folded localized excitation. Firstly, with the help of the Hirota’s bilinear method and ansatz, some periodic solutions have been derived. Secondly, taking Burgers equation as an auxiliary function, we have obtained n-soliton solution and n-shock wave. Lastly, we present a new variable separation method for (3+1)-dimensional and higher dimensional models, and use it to derive localized excitation solutions. To be specific, we have constructed various novel structures and discussed the interaction dynamics of folded solitary waves. Compared with the other methods, the variable separation solutions obtained in this paper not only directly give the analytical form of the solution u instead of its potential u_y, but also provide us a straightforward approach to construct localized excitation for higher order dimensional nonlinear partial differential equation.

Nonlinear radiation effects on water-based nanofluid containing CNTs subject to heat source/ sink past a wedge

In this paper, a two-dimensional and incompressible laminar flow comprised of water-based carbon nanotubes over convectively heated moving wedge under the magnetic field and nonlinear radiation and heat production/ absorption is investigated. The base nanofluid (water) contains single wall carbon nanotubes (SWCNTs) and multiple walls carbon nanotubes (MWCNTs). In order to convert the dimensional nonlinear partial differential equations in nondimensional nonlinear ordinary differential form, an adequate set of similarity variables had been used. These set of equations and boundary conditions are evaluated by the implementation of RKF-45 (Runge–Kutta–Fehlberg fourth-fifth) order scheme. The influence of several physical parameters on particular nanoparticle’s volume friction, temperature and velocity ratio parameter, heat source/ sink parameter, nonlinear radiative constraint, exponent constant, magnetic factor, Eckert and Biot numbers is studied. An opposite behavior of volume fraction and velocity ratio parameters on velocity and energy profiles is achieved.

An application of soliton solutions of a differential equation in the progression of aortic dissection

Aortic dissection is a serious pathology involving the vessel wall of the aorta with significant societal impact. To understand aortic dissection, we explain the role of the dynamic pathology in the absence or presence of structural and/or functional abnormalities. We frame a differential equation to evaluate the impact of mean blood pressure on the aortic wall and prove the existence and uniqueness of its solution for homeostatic recoil and relaxation for infinitesimal aortic tissue. We model and analyze a generalized (3 + 1)‐dimensional nonlinear partial differential equation for aortic wave dynamics. We use the Lie group of transformations on this nonlinear evolution equation to obtain invariant solutions, traveling wave solutions including solitons. We find that abnormalities in the dynamic pathology of aortic dissection act as triggers for the progression of disease in the early stage through the formation of soliton‐like pulses and their interaction. We address the role of unstable wavefields in waveform dynamics when waves are unidirectional. Moreover, the notion of dynamic pathology within the domain of vascular geometry may explain the evolution of aneurysms in cerebral arteries and cardiomyopathies even in the absence of anatomical and physiological abnormalities.

Simulations of the one and two dimensional nonlinear evolutionary partial differential equations: A numerical study

In this work a hybrid scheme is proposed for the numerical study of various evolutionary partial differential equations (EPDEs). In proposed strategy, temporal derivatives are estimated via first and second order finite differences while the solution and spatial derivatives are approximated by Lucas and Fibonacci polynomials. Further, the collocation approach is applied which convert the EPDEs, to the system of coupled linear equations which are simple to solve. For non-linear problems, Quasilinearization is used to tackle nonlinearity. The scheme is implemented to solve different EPDEs which include, one dimensional non-linear Boussinesq, Hunter–Saxton, wave-like and two dimensional linear and non-linear wave like equations. Accuracy of the scheme is portrayed by computing L∞,L2 error norms and the relative error. Moreover, the calculated outcomes are compared with previously existing results in literature. Simulation demonstrates that the scheme works well for the mentioned EPDEs.

Painlevé analysis, Bäcklund transformation and Exact solutions for the (3+1)-dimensional nonlinear partial differential equation represented by Burgers’ equation

Herein, the Painlevé analysis and Bäcklund transformation for the (3+1) dimensional Burger equation are presented. Using this analysis, it is shown that the equation under consideration non-integrable. But, it is under a constraint equation may be integrable. We construct the Bäcklund transformation for that equation. Similarity solutions for the mentioned equation have been obtained. Some of these solutions are completely new.

Multi-domain multivariate spectral collocation method for (2+1) dimensional nonlinear partial differential equations

The novelty of this work rests upon the use of the domain partitioning technique in time variable when discretizing the domain of solution in spectral collocation algorithm. The single domain multivariate spectral collocation based methods have been proven to be effective in solving time-dependent partial differential equations (PDEs) defined over small time domains. However, there is a significant loss of accuracy as time computational domain proliferates and also when the number of grid nodes approaches a definite particular number. Therefore, the establishment of the new innovative multi-domain multivariate spectral quasilinearisation method (MDMV-SQLM) is described for the purpose of solving (2+1) dimensional nonlinear PDEs defined on large time intervals. The main output of this study is confirmation that minimizing the size of time computational domain at each subinterval assures sufficiently accurate results that are attained using minimal number of nodal points and less computational time. The solution algorithm involves partitioning the time domain into multiple non-overlapping sub-domains, simplification of the nonlinear PDEs using the quasilinearisation method and assumption of approximate solutions using triple Lagrange interpolating polynomials with Chebyshev–Gauss–Lobatto (CGL) points. The multi-domain spectral collocation procedure is executed on the linear systems of algebraic equations, where the subsequent matrix systems are solved separately in every time sub-interval with the continuity equation essentially used in obtaining initial conditions in the next subintervals. MATLAB software is used to implement the solution algorithm and numerical results are demonstrated graphically and in tabular form. To highlight the efficaciousness and accuracy of the MDMV-SQLM, error estimates, condition numbers and computational time are presented for well known (2+1) dimensional nonlinear initial-Dirichlet boundary value problems. The adoption of the domain decomposition technique is efficacious in suppressing the numerical challenges linked to large matrices and ill-conditioned nature of the resulting coefficient matrix. Also, the communicated results confirm that the numerical scheme is computationally cheap, fast and yield extremely accurate and stable results with the aid of fewer number of grid points for large time domains.

Exact solutions of (1+2)-dimensional non-linear time-space fractional PDEs

PurposeIn this paper, the author presents a hybrid method along with its error analysis to solve (1+2)-dimensional non-linear time-space fractional partial differential equations (FPDEs).Design/methodology/approachThe proposed method is a combination of Sumudu transform and a semi-analytc technique Daftardar-Gejji and Jafari method (DGJM).FindingsThe author solves various non-trivial examples using the proposed method. Moreover, the author obtained the solutions either in exact form or in a series that converges to a closed-form solution. The proposed method is a very good tool to solve this type of equations.Originality/valueThe present work is original. To the best of the author's knowledge, this work is not done by anyone in the literature.

A novel and efficient method for obtaining Hirota’s bilinear form for the nonlinear evolution equation in (n+1) dimensions

Bilinearization of nonlinear partial differential equations (PDEs) is essential in the Hirota method, which is a widely used and robust mathematical tool for finding soliton solutions of nonlinear PDEs in a variety of fields, including nonlinear dynamics, mathematical physics, and engineering sciences. We present a novel systematic computational approach for determining the bilinear form of a class of nonlinear PDEs in this article. It can be easily implemented in symbolic system software like Mathematica, Matlab, and Maple because of its simplicity. The proven results are obtained by using a developed method in Mathematica and applying a logarithmic transformation to the dependent variable. Finally, the findings validate the implemented technique’s competence, productivity, and dependability. The approach is a useful, authentic, and simple mathematical tool for calculating multiple soliton solutions to nonlinear evolution equations encountered in nonlinear sciences, plasma physics, ocean engineering, applied mathematics, and fluid dynamics. • This paper introduces a new systematic computational approach for constructing bilinear forms for nonlinear PDEs. • The given approach can handle a class of (n+1)-dimensional nonlinear PDEs. • Various examples of well-known nonlinear evolution equations up to the (4+1)-dimension are shown. • The results of various nonlinear equations are provided using the proposed method.

Lump solutions to a generalized nonlinear PDE with four fourth-order terms

Abstract A combined fourth-order (2 + 1)-dimensional nonlinear partial differential equation which contains four fourth-order nonlinear terms and all second-order linear terms is formulated. This equation covers three generalized KP, Hirota–Satsuma–Ito, and Calogero–Bogoyavlenskii–Schiff equations as examples, which have physical applications in the study of various nonlinear phenomena in nature. In terms of some settings of the coefficients, a class of lump solutions is constructed by the Hirota bilinear method and the solutions are calculated through the symbolic computation system of Maple. Meanwhile, the relation between the coefficients and the solution is explored. Two special lump solutions are generated by taking proper values for the involved coefficients and parameters, and their dynamic behaviors are studied, as illustrative examples. The primary advantage of the Hirota bilinear method is to transform a nonlinear equation into a bilinear one so that the targeted equation can be easily studied.

Invariant subspace method for [formula omitted]-dimensional non-linear time-fractional partial differential equations

In this paper, we generalize the theory of the invariant subspace method to (m+1)-dimensional non-linear time-fractional partial differential equations for the first time. More specifically, the applicability and efficacy of the method have been systematically investigated through the (3+1)-dimensional generalized non-linear time-fractional diffusion–convection-wave equation along with appropriate initial conditions. This systematic investigation provides an important technique for finding a large class of various types of invariant subspaces with different dimensions for the above-mentioned equation. Additionally, we have shown that the obtained invariant subspaces help to derive a variety of exact solutions that can be expressed as the combinations of exponential, trigonometric, polynomial and well-known Mittag-Leffler functions.

Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

Unsteady boundary layer flow of Williamson nanofluids over a heated permeable stretching sheet embedded in porous medium in the presence of viscous dissipation

The main objective of this paper is to focus on a numerical study of unsteady boundary layer flow of Williamson Nanofluids over a heated permeable stretching sheet embedded in porous medium in the presence of viscous dissipation. A mathematical modeled which resembles the physical flow problem has been developed. By using an appropriate transformation, we converted the system of dimensional nonlinear partial differential equations into system of coupled dimensionless ordinary differential equations. Numerical solutions of these equations are obtained by Runge-Kutta fourth order with shooting method. The velocity, temperature and concentration distributions are discussed numerically and presented through graphs. The numerical values of reduced skin-friction coefficient, Nusselt number and Sherwood number at the plate are derived and discussed numerically for various values of physical parameters which are presented through tables. The present results have been compared with existing one for some limiting case and found excellent validation. It is analyzed that the reduced skin friction coefficient enhances with increasing values of an unsteady parameter, magnetic parameter and porosity parameter. In addition, we observe that decrement in velocity profile of the fluid flow is observed for increasing values of the non-Newtonian Williamson parameter and a rise in Eckert number leads to the enhancement of the temperature of the fluid in the thermal boundary layer.

A new set and new relations of multiple soliton solutions of (2 + 1)-dimensional Sawada–Kotera equation

A new transformation v=4(lnf)xx that can formulate a quintic linear equation and a pair of Hirota’s bilinear equations for the (2 + 1)-dimensional Sawada–Kotera (2DSK) Eq. (1) or u=4(lnf)x for 2DSK Eq. (2) is reported firstly, which enables one to obtain a new set of multiple soliton solutions of the 2DSK equation. They are not special cases of the known multiple solitons. The results presented in this paper show that the 2DSK equation not only possesses two sets of multiple soliton solutions, but also has a relation between them that the square of fn in n-soliton solution u=4(lnfn)x can also be obtained from the singularity limit of f2n in 2n-soliton solution u=2(lnf2n)x by using dual combination rules and a singular limit method. This property is unusual and the 2DSK equation is the first and only one found so far by us. Also, it establishes a connection of two equations because the quintic linear equation is solved by a pair of Hirota’s bilinear equations, of which one is the (2 + 1)-dimensional bilinear SK equation obtained under the case u=2(lnf)x, and the other is the bilinear KdV equation. The (1 + 1)-dimensional SK equation does not possess this property. As another example, a (3 + 1)-dimensional nonlinear partial differential equation possessing a pair of Hirota’s bilinear equations, however only bearing one set of multiple soliton solutions is studied.

Approximations of Lyapunov functionals for ISS analysis of a class of higher dimensional nonlinear parabolic PDEs

This paper introduces a novel method, namely the approximations of Lyapunov functionals, for input-to-state stability (ISS) analysis of a class of higher dimensional nonlinear parabolic partial differential equations (PDEs) with variable coefficients. Specifically, for any q∈[1,+∞] and the considered nonlinear parabolic PDEs with different types of boundary disturbances in Llocq(R+;L1(∂Ω)) and initial data in L1(Ω), we show that ISS-like estimates in L1-norm (or weighted L1-norm) can be established by constructing approximations of (coercive and non-coercive) Lyapunov functionals. Some examples are provided to illustrate the application of the proposed method.

Well posedness for a singular two dimensional fractional initial boundary value problem with Bessel operator involving boundary integral conditions

<abstract> This paper studies the existence and uniqueness of solutions of a non local initial boundary value problem of a singular two dimensional nonlinear fractional order partial differential equation involving the Caputo fractional derivative by employing the functional analysis. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of obtaining the a priori bound relies on the construction of a suitable multiplicator. From the resulted a priori estimate, we can establish the solvability of the associated linear problem. Then, by applying an iterative process based on the obtained results for the associated linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the considered nonlinear problem. </abstract>

Flow of Jeffrey Fluid over a Horizontal Circular Cylinder with Suspended Nanoparticles and Viscous Dissipation Effect: Buongiorno Model

Mathematical model of Jeffrey fluid describes the property of viscoelastic that clarifies the two components of relaxation and retardation times. Nevertheless, the poor thermal performance of Jeffrey fluid has been a key issue facing the public. This issue can be accomplished by the use of nanofluid that has superior thermal performance than the conventional fluids. A better cooling rate in industry is in fact not appropriate to attain by the thermal conductivity of the conventional fluids. On that account, the present study aims to delve into the impact of viscous dissipation and suspended nanoparticles on mixed convection flow of Jeffrey fluid from a horizontal circular cylinder. A concise enlightenment on the separation of boundary layer flow is included and discussed starting from the lower stagnation point flow up to the separation point only. The non-dimensional and non-similarity transformation variables are implemented to transform the dimensional nonlinear partial differential equations (PDEs) into two nonlinear PDEs, and then tackled numerically through the Keller-box method. Representation of tabular and graphical results are executed for velocity and temperature profiles as well as the reduced skin friction coefficient, Nusselt number and Sherwood number to investigate the physical insight of emerging parameters. It was found that the incremented ratio of relaxation to retardation, Deborah number and Eckert number have delayed the boundary layer separation up to 120o

Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general since they contain elliptic functions. Finally, we derive the conserved quantities of this equation by employing two approaches—the general multiplier approach and Ibragimov’s theorem. The importance of conservation laws is explained in the introduction. It should be pointed out that the investigation of higher dimensional nonlinear partial differential equations is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.

Nonlinear mixed convective nanofluid flow along moving vertical rough plate

The objective of the current research paper is to investigate the effects of surface roughness on magnetohydrodynamic nonlinear mixed convection nanofluid flow over vertically moving plate. The highly coupled dimensional nonlinear partial differential equations (NPDE) are converted into dimensionless NPDE along with the boundary conditions with the help of non-similar transformations. The resulting set of dimensionless nonlinear PDE’s are solved by using Quasilinearization technique and implicit finite difference method. Impacts of various dimensionless parameters, namely, Brownian diffusion (Nb), nonlinear mixed convection ( ), nanoparticle buoyancy ratio (Nr), Lewis number (Le), thermophoresis (Nt), frequency (n), magnetic (M) and small parameter ( ) are studied in detail on profiles as well as gradients. The results reveal that increasing values of increase the velocity profile, while increasing values of Nr decrease the same. Further, increasing values of and n exhibit sinusoidal variations on skin-friction coefficient, heat and nanoparticle mass transfer rates. Moreover, the presence of nonlinear mixed convection parameter has significant effects on fluid flow compared to its absence. In addition to this, rate of heat transfer is analyzed in presence and absence of nanoparticles.

Mathematical Solution on MHD Stagnation Point Flow of Ferrofluid

This study presents a numerical investigation on the magnetohydrodynamic (MHD) stagnation point flow of a ferrofluid with Newtonian heating. The black oxide of iron, magnetite (Fe3O4) which acts as magnetic materials and water as a base fluid are considered. The two dimensional stagnation point flow of cold ferrofluid against a hot wall under the influence of the uniform magnetic field of strength is located some distance behind the stagnation point. The effect of magnetic and volume fraction on the velocity and temperature boundary layer profiles are obtained through the formulated governing equations. The governing equations which are in the form of dimensional non-linear partial differential equations are reduced to dimensionless non-linear ordinary differential equations by using appropriate similarity transformation. Then, they are solved numerically by using the Keller-box method which is programmed in the Matlab software. It is found that the cold fluid moves towards the magnetic source that is close to the hot wall. Hence, leads to the better cooling rate and enhances the heat transfer rate. Meanwhile, an increase of the magnetite nanoparticles volume fraction, increases the ferrofluid capabilities in thermal conductivity and consequently enhances the heat transfer.