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In this paper, we introduce a new nonlinear evolution equation, which is ([Formula: see text])-dimensional variable-coefficient Kadomtsev–Petviashvili equation. First, according to the Hirota bilinear method, we get some exact solutions of the equation, including lump solution, lump-soliton solution, rogue-soliton solution and lump-kink solution. Then, we obtain some new exact solutions by generalizing the form of the lump solution on a further solution. Finally, based on the symbolic calculation method with Mathematica, the characteristics of the interaction solutions are shown in the graphs and we analyze the dynamic change of the solutions. Furthermore, we discuss the applications of these solutions in physics via the analysis.

The multiple rogue wave solutions method is employed for searching the multiple soliton solutions for the new variable-coefficient Kadomtsev-Petviashvili equation, which contains first-order, second-order, third-order, and fourth-order wave solutions. At the critical point, the second-order derivative and Hessian matrix for only one point will be investigated and the lump solution has one maximum value. The physical phenomena of these gained multiple soliton solutions are analysed and indicated in figures by selecting suitable values.

The main purpose of this study is to clarify the Hyers–Ulam stability (HUS) for the Cayley quantum equation. In addition, the result obtained for all parameters is applied to HUS of h-difference equations with a specific variable coefficient using a new transformation.

A system of boundary-domain integral equations is derived from the bidimensional Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix different from the o...

A fully-discrete approximate scheme is established for the 2D multi-term time-fractional mixed diffusion and diffusion-wave equations with spatial variable coefficient by using linear triangle finite element method in space and classical L 1 time-stepping method combined with Crank–Nicolson scheme in time. Then, the unconditionally stable analysis of the fully-discrete scheme is presented by employing some important lemmas. At the same time, both the spatial superclose property in H 1 -norm and convergence result in L 2 -norm are derived by skillfully dealing with numerical errors without any restrictions of time step τ and mesh size h . As a necessary way for obtaining the aimed numerical analysis, the relationship between the nonstandard projection operator R h and the interpolation operator I h of linear triangle finite element is introduced. Moreover, the global superconvergence is deduced by adopting interpolation postprocessing technique. Finally, numerical tests are provided to illustrate the efficiency and correctness of the theoretical results.

In this paper, a general propagation lattice Boltzmann model for a variable-coefficient compound Korteweg-de Vries-Burgers (vc-cKdVB) equation is investigated through selecting equilibrium distribution function and adding a compensation function, which can provide some more realistic models than their constant-coefficient counterparts in fluids or plasmas. Chapman–Enskog analysis shows that the vc-gKdVB equation can be recovered correctly from the present model. Numerical simulations in different situations of this equation are conducted, including the propagation and interaction of the bell-type, kink-type and periodic-depression solitons and the evolution of the shock-wave solutions. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current lattice Boltzmann model is a satisfactory and efficient algorithm. In addition, it is also shown the present model could be more stable and more accurate than the standard lattice Bhatnagar–Gross–Krook model through adjusting the two free parameters introduced into the propagation step.

Based on the spatial quasi-Wilson nonconforming finite element method and temporal L 2 − 1 σ formula, a fully-discrete approximate scheme is proposed for a two-dimensional time-fractional diffusion equations with variable coefficient on anisotropic meshes. In order to demonstrate the stable analysis and error estimates, several lemmas are provided, which focus on high accuracy about projection and superclose estimate between the interpolation and projection. Unconditionally stable analysis are derived in L 2 -norm and broken H 1 -norm. Moreover, convergence result of accuracy O ( h 2 + τ 2 ) and superclose property of accuracy O ( h 2 + τ 2 ) are deduced by combining interpolation with projection, where h and τ are the step sizes in space and time, respectively. And then, the global superconvergence is presented by employing interpolation postprocessing operator. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.

A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and delayed time reversal is derived. Various exact solutions of the VCmKdV equation, including elliptic periodic waves, solitary waves and interaction solutions between solitons and periodic waves are obtained and analyzed graphically. As an illustration, an approximate solution of the original nonlocal two-layer fluid system is also given.

To solve the problem of inaccurate position estimation due to the influence of magnetic saturation in sensorless control of switched reluctance motor (SRM), a new position estimation method is proposed. By analysing the linearity of the inductance curve, the region with better linearity can be found. Then, in this region, the mathematical model between rotor position and inductance can be built to estimate the rotor position by the linear fitting. However, the inductance curve will shift due to the influence of magnetic saturation in the case of heavy load, which makes the estimation error larger. Therefore, this study carries out a linear fitting of this region under different loads and establishes a linear inductance model with variable coefficient. This method can effectively eliminate the influence of saturation inductance on position estimation and have better estimation accuracy. Simulation and experimental results show the effectiveness and correctness of the proposed method.

In this paper, we present L 2 and negative-order norm estimates for the local discontinuous Galerkin (LDG) method solving variable coefficient Schrodinger equations. For these special solutions the LDG method exhibits “hidden accuracy”, and we are able to extract it through the use of a convolution kernel that is composed of a linear combination of B-splines. This technical was initially introduced by Cockburn, Luskin, Shu, and Suli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving (SIAC) filter. We demonstrate that it is possible to extend the SIAC filter on Schrodinger equations. When polynomials of degree k are used, we can prove theoretically the LDG method solutions are of order k + 1 , whereas the post-processed solutions that convolution with the SIAC filter are of order at least 2 k . Additionally, we present numerical results to confirm that the accuracy of LDG solutions can be improved from O ( h k + 1 ) to at least O ( h 2 k + 1 ) for Schrodinger equations by using alternating numerical fluxes.

The goal of this paper is to discuss high accuracy analysis of a fully-discrete scheme for 2D multi-term time fractional wave equations with variable coefficient on anisotropic meshes by approximat...

Propagation of nonlinear shock structures in inhomogeneous dusty plasmas with power law dust size distribution and nonadiabatic dust charge fluctuation has been investigated. Using reductive perturbation expansion, the hydrodynamic equations are reduced to a modified variable coefficient Korteweg–de Vries (MKdV)–Burgers equation governing the characteristic of shock structures. Two different types of shock structures exist in this system and transform each other. The numerical results show that the inhomogeneity, the power law dust size distribution and the nonadiabatic dust charge fluctuation have strong influence on the nonlinear properties of shock structures and the transition condition between monotone shock structures and oscillatory shock structures.

In this paper, we study a Cauchy problem of parabolic equations with time-dependent coefficients. The critical Fujita exponents are determined by the exponents of components and dimension of domain...

The development of modeling and predicting grinding force and energy has great significance in improving tool life and grinding efficiency. As an important parameter to study the grinding mechanism, the undeformed chip thickness catches more and more attentions in modeling the grinding process, and the investigation on its nonuniformity still has much potential. In the current work, the surface topology measuring and modeling methodologies were described. Models of the specific grinding force and energy were developed. Grinding experiments were conducted to investigate the effects of undeformed chip thickness nonuniformity. It is found that the specific grinding forces increase with increased mean value, decreased standard deviation (or variable coefficient) and increased percentage of active grains. The specific grinding energy increases with decreased mean value, increased standard deviation (or variable coefficient) and increased percentage of active grains. The undeformed chip thickness nonuniformity has a certain effect on the specific grinding forces and energy but less significant than the percentage of active grains. Finally, the tool life and grinding efficiency are improved according to the developed models with acceptable errors.

This study aims to investigate the problem of determining the unknown initial temperature in a variable coefficient heat equation. We obtain the existence and uniqueness of the solution of the optimal control problem considered under some conditions. Using the adjoint problem approach, we get the Frechet differential of the cost functional. We construct a minimizing sequence and give the convergence rate of this sequence. Also, we test the theoretical results in a numerical example by using the MAPLE® program.

In this paper, a general propagation lattice Boltzmann model for variable-coefficient non-isospectral Korteweg–de Vries (vc-nKdV) equation, which can describe the interfacial waves in a two layer liquid and Alfven waves in a collisionless plasma, is proposed by selecting appropriate equilibrium distribution function and adding the compensate function. The Chapman–Enskog analysis shows that the vc-nKdV equation can be recovered correctly from the present model. Numerical simulation for the non-propagating one soliton of this equation in different situations is conducted as validation. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current general propagation lattice Boltzmann model is a satisfactory and efficient method, and could be more stable and accurate than the standard lattice Bhatnagar–Gross–Krook model.

This study knows wheter or not there is an effect of giving incentives to employee performance. Type of quantitative research with an associative approach. The research conducted in PT.Telesindoshop Sorong city. The sample us of 65 people. As for the data sources used are primary data and secondary data. Data analysis uses simple linear regression with Statistical Product and Service Solution (SPSS). Based on the results of this study indicate that the provision of PT.Telesindo’s incentives has an influence on employee performance and significantly influence. Based on the results of data analysis, the incentive variable coefficient (X) is 0.564 while the constant is 28,282, so the regression equation can be obtained as follows: Y = 28,282 + 0,564 X. The provision of incentives influences the performance of PT. Telesindoshop employees. The results of the analysis indicate that the provision of incentives has a positive and significant effect on the performance of PT. Tesindoshop employees. With a significant value of 0.042, in this study used a significant level is 0.05 or 5% means 0.042

In this paper, the nonlocal symmetries and exact interaction solutions of the variable coefficient Korteweg–de Vries (KdV) equation are studied. With the help of pseudo-potential, we construct the high order nonlocal symmetries of the time-dependent coefficient KdV equation for the first time. In order to construct the new exact interaction solutions, two auxiliary variables are introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of the closed system, some exact interaction solutions are obtained. For some interesting solutions, such as the soliton–cnoidal wave solutions are discussed in detail, and the corresponding 2D and 3D figures are given to illustrate their dynamic behavior.

In this paper, we discuss the existence and uniqueness of solutions for a new kind of Langevin equation involving Riemann-Liouville as well as Caputo fractional derivatives, and variable coefficient, supplemented with nonlocal-terminal fractional integro-differential conditions. The proposed study is based on modern tools of functional analysis. We also extend our discussion to the associated inclusions problem. For the applicability of the obtained results, several examples are constructed. Some interesting observations are also presented.

In this paper, the inverse scattering transform is extended to a super Korteweg-de Vries equation with an arbitrary variable coefficient by using Kulish and Zeitlin?s approach. As a result, exact solutions of the super Korteweg-de Vries equation are obtained. In the case of reflectionless potentials, the obtained exact solutions are reduced to soliton solutions. More importantly, based on the obtained results, an approach to extending the scattering transform is proposed for the supersymmetric Korteweg-de Vries equation in the 1-D Grassmann algebra. It is shown the the approach can be applied to some other supersymmetric non-linear evolution equations in fluids.

This paper is concerned with the numerical simulations for a class of variable coefficient fractional Burgers equations with delay. The proposed numerical scheme has lower computational cost than the traditional implicit difference schemes and can recover some integrable properties of the original equations. The boundedness, unconditional stability, and the global convergence of the scheme are proved. A numerical experiment is provided to confirm the theoretical results.

In this paper, the improved CK reduction transformation is performed on the generalized variable-coefficient KdV (vc-KdV) type of equation, then these variable-coefficient equations are transformed into its constant-coefficient counterparts under some conditions. Moreover, the complete Lie group classification is presented, all of the point symmetries of the equations are obtained. Furthermore, the exact solutions and conservation laws of the equations are investigated.

In order to study steamed bread quality traits and explore good wheat cultivars, 25 wheat cultivars grown in Shandong province were used through one-process fermentation procedure to examine the variability of steamed bread quality traits and the correlation between quality and score. The results showed that all the steamed bread quality traits expressed large variable coefficient except volume, L* value, l* value and cohesiveness traits, which suggested that the genetic diversity made from wheat in Shandong province was large. The average comprehensive score of the steamed bread was 76.8 for the tested 25 varieties, and ten varieties reached the good steamed bread level, which were Liangxing 99, Luyuan 205, Jimai 22, Jinan 17, Tainong 19, Wennong 17, Huaiyin 84137, and Huaimai 8. The main quality factors affecting the steamed bread score were volume, L* value, A* value, hardness, chewiness, and adhesiveness. Among these factors, the adhesiveness has a significantly positive correlation with the hardness and cohesive, and the volume has a significantly positive correlation with the A* value. The volume, L* value has a significantly positive correlation with score, A* value, hardness and chewiness, and gumminess has a significantly negative correlation with the score. There was no correlation between the score and other traits. The results will benefit the understanding of steamed bread quality in wheat cultivated in Shandong. l

Objectives: The aim is to study the application of artificial bee colony (ABC) algorithm in medical image threshold segmentation. Methods: A new image segmentation method based on the improved ABC and thresholding medical image threshold segmentation method is proposed, which is variable coefficient ABC (VCABC) optimization algorithm, which is used to determine n-1 optimal n level threshold on a given image. The proposed method is compared with the Particle Swarm Optimization fractional image threshold segmentation method and the ABC fractional medical image threshold segmentation method. Results: When considering a variety of conditions, the performance of this method is better than that of other methods. Conclusions: The improved method of combining ABC and fractional medical image threshold segmentation method is effective.

Under investigation in this paper is a new and more general non-isospectral and variable-coefficient non-linear integrodifferential system. Such a system is Lax integrable because of its derivation from the compatibility condition of a generalized linear non-isospectral problem and its accompanied time evolution equation which is generalized in this paper by embedding four arbitrary smooth enough functions. Soliton solutions of the derived system are obtained in the framework of the inverse scattering transform method with a time-varying spectral parameter. It is graphically shown the dynamical evolutions of the obtained soliton solutions possess time-varying amplitudes and that the inelastic collisions can happen between two-soliton solutions.

Motivated by the smoothed particle hydrodynamics (SPH), we present nonlocal models for linear advection with a variable coefficient in one spatial dimension together with their particle based numer...

We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type doe...

We investigate some well-known variable-coefficient time-fractional nonlinear systems of partial differential equations using the Lie symmetry method and derive their symmetries and reductions into fractional nonlinear systems of ordinary differential equations.

In this paper, we study some exact solutions of the (3 + 1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles. Based on a given transformation and the bilinear form, together with symbolic computations, we obtain the multi-soliton solutions and periodic solutions including X-periodic, Y-periodic and 2-periodic wave solutions. By using of the long wave limit method, we obtain lump solutions and lump-periodic solutions that lumps moving on a background with different periodic waves. The properties of the solutions are analyzed graphically.

In this paper, we consider a variable coefficient thermo-viscoelastic-coupled system of second sound with acoustic boundary conditions, where the heat conduction is given by Cattaneo’s law. We establish a general decay of energy-associated solutions under a class of generality of the relaxation function. Our result extends the various decay results obtained for problems with or without thermo-viscoelasticity.