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The (2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis (CD) and fractional potential Kadomstev-Petviashvili (PKP) equation. It can be modeled according to the Hamiltonian structure, the lax pair with the non-isospectral problem, and the pain level property. The proposed equations are widely used in beachfront ocean and coastal engineering to describe the propagation of shallow-water waves, demonstrate the propagation of waves in dissipative and nonlinear media, and reveal the propagation of waves in dissipative and nonlinear media. In this paper, we have established further exact solutions to the nonlinear fractional partial differential equation (NLFPDEs), namely the space-time fractional CD and fractional PKP equations using the modified Rieman-Liouville fractional derivative of Jumarie through the two variable ( G ′ / G , 1 / G )-expansion method. As far as trigonometric, hyperbolic, and rational function solutions containing parameters are concerned, solutions are acquired when unique characteristics are assigned to the parameters. Subsequently, the solitary wave solutions are generated from the solutions of the traveling wave. It is important to observe that this method is a realistic, convenient, well-organized, and ground-breaking strategy for solving various types of NLFPDEs.

In this article, a (2+1)-dimensional Korteweg-de Vries equation is investigated. Abundant periodic wave solutions are obtained based on the Hirota’s bilinear form and a direct test function. Meanwhile, the interaction solutions between lump and periodic waves are presented. What is more, we derive the interaction solutions among lump, periodic, and solitary waves. Based on the extended homoclinic test technique, some new double periodic-soliton solutions are presented. Finally, some 3D and density plots are simulated and displayed to respond the dynamic behavior of these obtained solutions.

The space–time fractional equal width (EW) and the space–time fractional generalized equal width (GEW) equations are two important models that represent nonlinear dispersive waves, namely, waves ensuing in the shallow water channel, 1-D wave generation ascending in the nonlinear dispersive medium estimation, cold plasma hydro-magnetic waves, chemical kinematics, electromagnetic interaction, etc. In this article, we search advanced and broad-spectrum wave solutions of the formerly indicated models in diverse families in conjunction with the Riemann-Liouville fractional derivative via the double ( G ′ / G , 1 / G )-expansion approach. The nonlinear fractional differential equations (NLFDEs) are transformed into ODEs by the composite function derivative and the chain rule putting together with the wave transformation. We acquire kink wave solution, multiple periodic solutions, single soliton solution, periodic wave solution and other types of soliton solutions by setting particular values of the embodied parameters. The suggested technique is functional, convenient, powerful, and computationally feasible to examine scores of NLFDEs.

This work aims to construct exact solutions for the space-time fractional (2+1)- dimensional dispersive longwave (DLW) equation and approximate long water wave equation (ALW) utilizing the two-variable ( G ′ / G , 1 / G ) -expansion method and the modified Riemann–Liouville fractional derivative. The recommended equations play a significant role to describe the travel of the shallow water wave. The fractional complex transform is used to convert fractional differential equations into ordinary differential equations. Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package. The Maple package program was used to set up and validate all of the computations in this investigation. By choosing particular values of the embedded parameters, we produce multiple periodic solutions, periodic wave solutions, single soliton solutions, kink wave solutions, and more forms of soliton solutions. The achieved solutions might be useful to comprehend nonlinear phenomena. It is worth noting that the implemented method for solving nonlinear fractional partial differential equations (NLFPDEs) is efficient, and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

Nonlinear fractional partial differential equations are highly applicable for representing a wide variety of features in engineering and research, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. In this research, we chose to construct some new closed form solutions of traveling wave of fractional order nonlinear coupled type Boussinesq–Burger (BB) and coupled type Boussinesq equations. In beachside ocean and coastal engineering, the suggested equations are frequently used to explain the spread of shallow-water waves, depict the propagation of waves through dissipative and nonlinear media, and appears during the investigation of the flow of fluid within a dynamic system. The subsidiary extended tanh-function technique for the suggested equations is solved for achieve new results by conformable derivatives. The fractional order differential transform was used to simplify the solution process by converting fractional differential equations to ordinary type differential equations by using the mentioned method. Using this technique, some applicable wave forms of solitons like bell type, kink type, singular kink, multiple kink, periodic wave, and many other types solution were accomplished, and we express our achieve solutions by 3D, contour, list point, and vector plots by using mathematical software such as MATHEMATICA to express the physical sketch much more clearly. Moreover, we assured that the suggested technique is more reliable, pragmatic, and dependable, that also explore more general exact solutions of close form traveling waves.

We opted to construct a traveling wave solution to the nonlinear space-time fractional coupled modified equal width (CMEW) equation and the space-time fractional-coupled Burgers equation, which are often used as an electro-hydro-dynamical model to advance the local electric field and particle acoustic waves in plasma, the shallow water wave issues and portray the variety additional time of an actual structure on the partial liquid mechanics framework, particle acoustic waves through a gas-filled line, and certain consistent state gooey liquid. In this study, we employ the two variable (G /G, 1/G)-expansion method to create further general solitary wave solutions to those equations based on Riemann– Liouville fractional derivative. The fractional differential wave transform simplifies by generating ordinary differential equations (ODE) from fractional-order differential equations. We identified multiple types of solutions through the maple that are illustrated using 3D shape, 3D list point plot, and contour narratives. Additionally, we proposed that the methodology be changed to be more pragmatic, economical, and dependable and that we investigate more generalized precise solutions for traveling waves, such as solitary wave solutions.

This research aims to use the shooting method (SM) to find numerical solutions to the boundary value problems of ordinary differential equations (ODEs). Applied mathematics, theoretical physics, engineering, control, and optimization theory all have two-point boundary value problems. If the two-point boundary value problem cannot be solved analytically, numerical approaches must be used. The scenario in the two-point boundary value issue for a single second-order differential equation with prescribed initial and final values of the solution gives rise to shooting method. Firstly, the method is discussed, and some boundary value problems of ODEs are solved by using the proposed method. Obtained results are compared with the exact solution for the validation of the proposed method and represented both in graphical and tabular form. It has been found that the convergence rate of the shooting method to the exact solution is so high. As a finding of this research, it has been determined that the shooting method produces the best-fit numerical results of boundary value problems.

The space-time fractional telegraph equation and the space-time fractional modified third-order Kdv equations are significant molding equations in theoretic physics, mathematical physics, plasma physics also other fields of nonlinear sciences. The space time-fractional telegraph equation, which appears in the investigation of an electrical communication line and includes voltage in addition to current which is dependent on distance and time, is also applied to communication lines of wholly frequencies, together with direct current, as well as high-frequency conductors, audio frequency (such as telephone lines), and low frequency (for example cable television) used in the extension of pressure waves into the lessons of pulsatory blood movement among arteries also the one-dimensional haphazard movement of bugs towards an obstacle. The presence of chain rule and the derivative of composite functions allows the nonlinear fractional differential equations (NLFDEs) to translate into the ordinary differential equation employing wave alteration. To explore such categories of resolutions, the extended tanh-method is accomplished via Conformable fractional derivatives. A power sequence in tanh was originally used as an ansatz to provide analytical solutions of the traveling wave type of certain nonlinear evolution equations. To convert this problem to a standard differential equation, a partial complex transformation that has been summarized succinctly is utilized correctly thus, with all of the free parameters, numerous exact logical arrangements are required. The results are found as hyperbolic and rational functions involving parameters, when specific values are supplied to the parameters solitary wave solutions are formed from traveling wave solutions. The outcomes achieved in this study are king type, single soliton, double soliton, multiple solitons, bell shape, and other sorts of forms and we demonstrated that these solutions were validated through the Maple software. The proposed approach for solving nonlinear fractional partial differential equations has been developed to be operative, unpretentious, quick, and reliable to usage.

We utilize the fractional modified Riemann-Liouville derivative in the sense to develop careful arrangements of space–time fractional coupled Boussinesq equation which emerges in genuine applications, for instance, nonlinear framework waves iron sound waves in plasma and in vibrations in nonlinear string and space–time fractional-coupled Boussinesq Burger equation that emerges in the investigation of liquids stream in a dynamic framework and depicts engendering of shallow-water waves. A decent comprehension of its solutions is exceptionally useful for beachfront and engineers to apply the nonlinear water wave model to the harbor and seaside plans. A summed-up partial complex transformation is correctly used to change this equation to a standard differential equation thus, many precise logical arrangements are acquired with all the free parameters. At this point, the traveling wave arrangements are articulated by hyperbolic functions, trigonometric functions, and rational functions, if these free parameters are considered as specific values. We obtain kink wave solution, periodic solutions, singular kink type solution, and anti-kink type solutions which are shown in 3D and contour plots. The presentation of the method is dependable and important and gives even more new broad accurate arrangements.

The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables G ′ / G , 1 / G -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of tanh , sech , sinh , rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.

The exact wave solutions to the space-time fractional modified Benjamin-Bona-Mahony (mBBM) and space time fractional Zakharov-Kuznetsov Benjamin-Bona-Mahony (ZKBBM) equations are studied in the sense of conformable derivative. The existence of chain rule and the derivative of composite functions permit the nonlinear fractional differential equations (NLFDEs) to convert into the ordinary differential equation using wave transformation. The wave solutions of these equations are examined by means of the expanding and effective two variable ( G' / G ,1/ G )-expansion method. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions containing parameters. The method is efficient, convenient, accessible and is the generalization of the original ( G' / G )-expansion method.

The nonlinear Riemann wave equations (RWEs) and the Landau-Ginsburg-Higgs (LGH) equation are related to plasma electrostatic waves, ion-cyclotron wave electrostatic potential, superconductivity, and drift coherent ion-cyclotron waves in centrifugally inhomogeneous plasma. In this article, the interactions between the maximum order linear and nonlinear factors are balanced to compute realistic soliton solutions to the formerly stated equations in terms of hyperbolic functions. The linear and nonlinear effects rheostat the structure of the wave profiles, which vary in response to changes in the subjective parameters combined with the solutions. The established solutions to the aforementioned models using the extended tanh scheme are descriptive, typical, and consistent, and include standard soliton shapes such as bright soliton, dark soliton, compacton, peakon, periodic, and others that can be used to analyze in ion-acoustic and magneto-sound waves in plasma, homogeneous, and stationary media, particularly in the propagation of tidal and tsunami waves.

The space time fractional modified regularized long wave equation is a model equation to the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, the hydro-magnetic waves in cold plasma, the phonetic waves in dissident quartz and phonetic gravitational waves in contractible liquids. In nonlinear science and engineering, the mentioned equation is applied to analyze the one way tract of long waves in seas and harbors. In this study, the closed form traveling wave solutions to the above equation are evaluated due to conformable fractional derivatives through double (G'⁄G,1⁄G)-expansion method and the Exp-function method. The existence of chain rule and the derivative of composite function permit the nonlinear fractional differential equations (NLFDEs) converted into ODEs using wave transformation. The obtain solutions are very much effective to analyze the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, one way tract of long waves in seas and harbors. These two methods are efficient, convenient, and computationally attractive.

The two variable (G'⁄G, 1⁄G)-expansion method is significant for finding the exact traveling wave solution to nonlinear evolution equations (NLEEs) in mathematical physics, applied mathematics and engineering. In this article, we exert the two variable (G'⁄G, 1⁄G)-expansion method for investigating the fractional generalized reaction Duffing model and density dependent fractional diffusion reaction equation and obtain exact solutions containing parameters. When the parameters are taken particular values, traveling wave solutions are transferred into the solitary wave solutions. The two variable (G'⁄G, 1⁄G)-expansion method is the generalization of the original (G'⁄G)-expansion method established by Wang et al [21].

Physico-chemical modifications of hydroxyapatite (HAp) materials are considered as pre-requisites for the development of new bioactive carrier materials for drug delivery and tissue engineering applications. Since acidic amino acids have well-documented affinities to both HAp and basic proteins, HAp modified by aspartic acid (Asp, acidic amino acid) might be one of the candidate substrates for a basic protein carrier. Here, we synthesized HAp in the presence of various concentrations of Asp and observed that HAp crystallinity and other physico-chemical properties were effectively modulated. Detailed studies indicated that Asp was not incorporated in the HAp crystal lattice, but rather was trapped in HAp crystals. Protein adsorption studies indicated that the HAp particles modified by Asp had a selective loading capacity for basic protein. Therefore, HAp particles containing Asp might have potential in drug delivery applications, especially as the carrier of basic proteins including bFGF and BMP.