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Based on three-wave method, abundant nonautonomous solutions with different wave structures of a 3D variable-coefficient generalized shallow water wave equation are presented. In this situation the basic configurations (characteristic, amplitude, and speed) of the nonautonomous waves mutate with time due to the effect of nonlinearity and dispersion. Discussions indicate the interaction between lump wave and a pair of resonance stripe solitons solutions, the interaction between lump wave and periodic wave solutions, and the breather-type periodic soliton solutions. The dynamic properties of the obtained solutions are demonstrated and analyzed graphically for different choices of the arbitrary functions in these solutions.

Cosmic plasmas are considered as the most abundant form of ordinary matter in the Universe while observations of the cosmic dust in different regions provide an insight into the Universe's recyclin...

In this study we shall obtain some sufficient conditions for the existence of positive solutions of variable coefficient nonlinear second-order neutral differential equation with distributed deviating arguments. For some different cases of the range of p(t) by using Banach contraction principle we will give some sufficient conditions for the nonoscillatory solutions of second-order neutral differential equation. With this purpose we will use fixpoint theorem. At the end of the research, there is an example that meets the conditions we have given. Our results improve and extend some existing results.

Under investigation is a generalized variable-coefficient Kadomtsev–Petviashvili equation with self-consistent sources. Our main job is divided into four parts: (i) lump wave solution, (ii) interaction solutions between lump and solitary wave, (iii) breather wave solution and (iv) interaction solutions between lump and periodic wave. Furthermore, the interaction phenomenon of waves is shown in some 3D- and contour plots.

This paper investigates explicit power series solutions and conservation laws of variable coefficient time fractional Kawahara equation with Riemann–Liouville derivative which describs oscillatory ...

The paper studies one combined mixed problem for the Klein–Gordon–Fock equation with a variable coefficient. The solvability of the problem under consideration is proved. In addition, it is established that the solution to the mixed problem under study is stable with respect to an additive perturbation of the coefficient, as well as with respect to the boundary conditions and the right-hand side of the equation.

We consider direct and inverse abstract Cauchy problems with time-dependent coefficients. We give an explicit representation of the unique solution operator of the direct abstract Cauchy problem. W...

The object of this fundamental research will be to reveal the effects of variable coefficient of static friction on the static behavior of disc spring. On account of the complex relationship between friction and normal pressure, based on contact mechanics and fractal theory, a model for predicting variable coefficient of static friction of 3 D fractal surface is induced. Then the method proposed by Curti is improved by using the fractal coefficient of static friction to replace the Coulomb's law. The entire deflection range is considered under quasi-static loading condition. The model of disc spring is developed to predict the load-deflection and hysteresis characteristics while contacting with different fractal surfaces. The primary object of this study is to survey the variable static friction coefficient and its effect on the disc spring. Different fractal dimension, constant static friction coefficient and free height of disc spring are considered, and the relative error of the two methods are calculated. The investigation has revealed that the effect of fractal static friction coefficient on the load-deflection and hysteresis characteristics of disc spring is concentrated in the small deflection stage. And the method is universal for disc springs of different dimension on the same surface.

Generalized plane waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz pro...

In this paper, a new integrable variable coefficient Toda equation is proposed by utilizing a generalized version of the dressing method. At the same time, we derive the Lax pair of the new integrable variable coefficient Toda equation. The compatibility condition is given, which insures that the new Toda equation is integrable. To further analyze the character of the Toda equation, we derive one soliton solution of the obtained Toda equation by using separation of variables.

基于Hirota双线性方法,利用拓展的同宿呼吸检验法得到了(3+1)维变系数Kudryashov-Sinelshchikov(K-S)方程的同宿呼吸波解,对该解的参数选取合适的数值,可得到不同结构的同宿呼吸波.通过对同宿呼吸波解的周期取极限,推导出方程的怪波解.最后,构造出一个特殊的高阶多项式作为测试函数,求得该方程的一阶怪波解和二阶怪波解.

Изучается вольтерровское интегро-дифференциальное уравнение типа свертки со степенной нелинейностью, переменным коэффициентом $a(x)$ и неоднородностью $f(x)$ в линейной части, которое тесно связано с соответствующим нелинейным интегральным уравнением, возникающим при исследовании инфильтрации жидкости из цилиндрического резервуара в изотропную однородную пористую среду, при описании процесса распространения ударных волн в трубах, наполненных газом, при решении задачи о нагревании полубесконечного тела при нелинейном теплопередаточном процессе, в моделях популяционной генетики и других. Важно отметить, что в связи с указанными и другими приложениями особый интерес представляют непрерывные положительные при $x>0$ решения интегрального уравнения. На основе полученных точных нижней и верхней априорных оценок решения интегрального уравнения мы строим весовое полное метрическое пространство $P_b$, инвариантное относительно нелинейного интегрального оператора свертки, порожденного этим уравнением, и, применяя метод весовых метрик (аналог метода Белицкого), доказываем глобальную теорему о существовании и единственности решения изучаемого нелинейного интегро-дифференциального уравнения как в пространстве $P_b$, так и во всем классе $Q_0^1$ непрерывно дифференцируемых положительных при $x>0$ функций. Показано, что решение может быть найдено в пространстве $P_b$ методом последовательных приближений пикаровского типа. Для последовательных приближений получены оценки скорости их сходимости к точному решению в терминах весовой метрики пространства~$P_b$. В частности, при $f(x)=0$ из этой теоремы вытекает, что соответствующее однородное нелинейное интегро-дифференциальное уравнение, в отличие от линейного случая, имеет нетривиальное решение. Приведены также примеры, иллюстрирующие полученные результаты.

El objetivo de este estudio fue evaluar el efecto calendario del mes sagrado de Ramadán sobre el volumen de operaciones y los rendimientos anormales en el mercado de capitales iraní durante los años 2009 a 2019 durante 10 años en la bolsa de valores. El presente estudio se considera aplicado en términos de propósito y correlación en términos de método descriptivo. Además, se utilizó el modelo de panel de datos para investigar la relación entre las variables. De acuerdo con los resultados, la variable coeficiente del efecto del mes sagrado de Ramadán sobre el volumen de negociación tiene un nivel significativo de 0.441, por lo tanto, considerando que este valor es mayor que el nivel alfa de la investigación, que es igual a 0.05, por lo tanto, no se rechaza esta hipótesis (cero) de que la variable del efecto del Ramadán no es efectiva sobre la variable de volumen de comercio. Por tanto, el efecto del calendario del mes sagrado sobre el volumen de transacciones no es significativo. Según los resultados la variable coeficiente del efecto del Ramadán y el retorno de la acción tiene un nivel significativo de 0.000, por lo que considerando que este valor es menor que el nivel alfa de la investigación, que es igual a 0.05, por lo tanto, esta hipótesis (cero) que la variable de efecto Ramadán D01 no afecta la rentabilidad de las acciones en Irán, se rechaza.

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

In this paper we consider uniform stabilization of the wave equation with variable coefficients in a bounded domain. The nonlinear damping is put partly on the interior of the domain and partly on ...

A fifth-order variable coefficient nonlinear Schrodinger equation based on original inhomogeneous model is proposed and studied. Bright multi-soliton analytic solutions of the equation are calculated through the Hirota method and auxiliary function. The effect of different constraints between fifth-order dispersion with third-order dispersion on soliton transmission is researched. Besides, their propagation and interaction dynamics are analyzed. Moreover, based on the obtained three-soliton solutions, we change the value of $$\beta (x)$$ to discuss the soliton propagation. Three different propagation and interaction structures are derived via adjusting the nonlinear term. The results can enrich the inhomogeneous model and apply in nonlinear optical fiber.

Under investigation is a generalized (3 + 1)-dimensional variable- coefficient Kadomtsev– Petviashvili equation in fluid mechanics. Various exact analytical solutions are obtained by Hirota’s bilinear method, such as lump-type, breather wave and kink-solitary wave solutions. We discuss the interaction between lump wave and solitary waves, and the interaction between lump wave and periodic wave. The physical structure and propagation characteristics of obtained solutions are shown by some 3D graphics.

In this paper, based on Alikhanov’s L2- $$1_\sigma $$ high-order approximation and anisotropic finite element methods, a fully discrete scheme for time fractional variable coefficient diffusion equations on anisotropic meshes is presented. Firstly, we prove that the discrete scheme is unconditionally stable in $$H^1$$ -norm, then the results of convergence in $$L_2$$ -norm and superclose in $$H^1$$ -norm are derived by combining interpolation with projection, and then, the superconvergence in $$H^1$$ -norm is obtained by using interpolation post-processing technique. In addition, it is worth mentioning the key technology of combining interpolation and projection. If interpolation or projection is used alone, the results of this article cannot be obtained. Finally, numerical examples are provided to verify the correctness of theoretical analysis.

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent., 27 pages, 3 figures. Update incorporates referee suggestions. Minor correction to the proof of Lemma 2.6. To appear in APDE

In this paper, Modified Variational Iteration Method (MVIM) and Homotopy Analysis Method (HAM) are used to find approximate solutions for the Variable-Coefficient Variant Boussinesq System the (VCVB) system is able to describe the nonlinear and dispersive long gravity waves in shallow water traveling in two horizontal directions with varying depth, as an example we took the Boussinesq-Burgers (B-B) system, (B-B) system arise in the study of fluid flow and describing the long-wave propagation of shallow water waves. The solutions of these equations helpful for the coastal and civil engineering’s

A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coefficients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coefficients as long as all the requirements of the laws of physics are satisfied. To check the validity and accuracy of the proposed numerical method, some specific test problems with known solutions are solved.

In this paper, we study a nonlinear diffusion-convection-reaction equation with a variable coefficient which has applications in many fields. The Lie point symmetries of this equation are derived, according to which this equation is classified into four different kinds. Conservation laws for this equation are constructed by using the conservation theorem of Ibragimov.

In this paper, through symbolic computations, we obtain two exact solitary wave solitons of the (2 + l)-dimensional variable-coefficient Caudrey-Dodd- Gibbon-Kotera-Sawada equation. We study basic properties of l-periodic solitary wave solution and interactional properties of 2-periodic solitary wave solution by using asymptotic analysis.

In this paper, we consider the uniform stabilization problem for a coupled 1-d variable coefficient wave equation with anti-dampings in the interior domain and joint difference-type control. We use...

In this paper, nonlocal symmetries of variable coefficient Ablowitz-Kaup-Newell-Segur(AKNS) system are studied for the first time. In order to construct some new analytic solutions, a new variable is introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of closed system, we give out two types of symmetry reductions and some analytic solutions. For some interesting solutions, such as interaction solutions among solitons and other complicated waves, we give corresponding images to describe their dynamic behavior.

We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein--Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubic...

A fast and accurate analytical inverter-fed induction machine iron loss calculation model is proposed in this paper. The proposed model takes account of the influence of the output voltage harmonics from the inverter on the iron loss of the motor based on the piecewise variable coefficients method. Additionally, the proposed method incorporates slot harmonic component's influence on the iron loss. The validity of the proposed model is verified by comparing its calculated core loss values with measured ones of a 5.5- and a 55-kW inverter-fed induction motors under different speeds and load conditions. Compared with the classical iron loss model and piecewise variable coefficient iron loss model based on the finite-element method, the proposed model can reduce the computational burden significantly with desirable accuracy.

Based on the generalized dressing method, we propose a new integrable variable coefficient Spin-1 Gross–Pitaevskii equations and derive their Lax pair. Using separation of variables, we have derived explicit solutions of the equations. In order to analyze the characteristic of derived solution, the graphical wave of the solutions is plotted with the aid of Matlab.

Today, most of the real physical world problems can be modeled with variable-coefficient KdV–modified KdV (vcKdV–mKdV) equation. Besides, the solution methods and their reliabilities are the most important. Therefore, a high precision numerical method is always needed. In this paper, Fourier spectral method is applied to solve the space fractional generalized vcKdV–mKdV equation and the influence of fractional orders on numerical solution of the space fractional generalized vcKdV–mKdV equation is investigated. Numerical simulations in different situations of equation are conducted, including the propagation and interaction of the generalized ball-type, kink-type and periodic-depression solitons. From the numerical experiments pondered here and compared with the other methods, it is found that the numerical solutions match well with the exact solutions, which demonstrate that the Fourier spectral method is a satisfactory and efficient algorithm.

We establish the Ulam stability of a first-order linear nonautonomous quantum equation with Cayley parameter in terms of the behavior of the nonautonomous coefficient function. We also provide details for some cases of Ulam instability.

Fractional-order calculus has become a useful mathematical framework to describe the complex super-diffusive process; however, numerical solutions of the two-sided space-fractional super-diffusive model with variable coefficients are difficult to obtain, and almost no method can obtain an analytical solution. In this paper, a class of new fractional dimensional reproducing kernel spaces (RKS) based on Caputo fractional derivatives is given, and we give analytical and numerical solutions of the two-sided space-fractional super-diffusive model based on the class of new RKS. The analytical solution is represented in the form of series in the reproducing kernel space. Numerical experiments indicate that the piecewise reproducing kernel method is more accurate than the traditional reproducing kernel method (RKM), and these new fractional reproducing kernel spaces are efficient for the two-sided space-fractional super-diffusive model.

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 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.