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Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f ′ ′ + P ( z ) f = 0 , where P ( z ) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E. Hille’s 1969 book Lectures on Ordinary Differential Equations are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille’s book.

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.

In the wake of the 2020 COVID-19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic, and of disease models generally. Most works follow the susceptible-infected-removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID-19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult-to-estimate, compartments (such as exposed individuals) to incorporate time delays. In the present work, we introduce a DDE epidemic model in both an ordinary- and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model’s ability to reproduce measured data on realistic problems.

Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on $D^n$-finiteness of solutions of first order differential equations., 19 pages

Following [C. Yu, Y. Wang, and D. Cheng, Tail behavior of the sums of dependent and heavy-tailed random variables, J. Korean Stat. Soc., 44(1):12–27, 2015], we revisit the asymptotic formulas of tail probabilities of randomly weighted sums and their maxima. Concretely, let {Xn, n ≥ 1} be a sequence of nonnegative random variables with common strong subexponential distribution, which are dependent according to a pretty weak structure, and let {θn, n ≥ 1} be another sequence of nonnegative and arbitrarily dependent random variables but independent of {Xn, n ≥ 1}. For any fixed n ≥ 1, under some mild conditions, we derive the asymptotics of the tail probability of randomly weighted sums $$ {S}_n^{\theta } $$ = $$ {\sum}_{i=1}^n{\theta}_i{X}_i $$ and their maxima $$ {M}_n^{\theta } $$ = max1≤m≥n $$ {S}_m^{\theta } $$ . Our result substantially extends some existing ones in the literature.

The motion of water conveying copper and aluminum nanoparticles on a heated moving sheet when thermal radiation and stretching/shrinking surface is significant and is investigated in this study to announce the increasing effects of volume fractions, thermal radiation, and moving parameters on this transport phenomenon. Furthermore, the flow of a Cu − Al 2 O 3 /water hybrid nanofluid across a heated moving sheet has been studied in both cross and streamwise directions. Thermal radiation effect is also considered, as this effect along with cross flow has not yet been investigated for the hybrid nanofluid in the published literature. Two distinct types of nanoparticles, namely, Al 2 O 3 (alumina) and Cu (copper), have been used to prepare hybrid nanofluid where water is considered as a base fluid. The system of nonlinear partial differential equations (PDEs) has been transferred to ordinary differential equations (ODEs) by compatible transformations before solving them by employing the III-stage Lobatto-IIIa method in bvp4c solver in MATLAB 2017 software. Temporal stability analysis has been carried out in order to verify stable branch between two branches by obtaining the smallest eigenvalue values. The branches obtained are addressed in depth against every applied parameter using figures and tables. The results show that there are three ranges of branches, no solution exists when λ > λ c , dual branches exist when 0.23 ≤ λ ≤ λ c , and a single solution exists when λ > 0.23 . Moreover, thermal layer thickness declines initially and then enhances in the upper and lower solutions for the higher values of the thermal radiation parameter.

This article is concerned with the fluid mechanics of MHD steady 2D flow of Williamson fluid over a nonlinear stretching curved surface in conjunction with homogeneous-heterogeneous reactions with convective boundary conditions. An effective similarity transformation is considered that switches the nonlinear partial differential equations riveted to ordinary differential equations. The governing nonlinear coupled differential equations are solved by using MATLAB bvp4c code. The physical features of nondimensional Williamson fluid parameter λ , power-law stretching index m , curvature parameter K , Schmidt number Sc , magnetic field parameter M , Prandtl number Pr , homogeneous reaction strength k 1 , heterogeneous reaction strength k 2 , and Biot number γ are presented through the graphs. The tabulated form of results is obtained for the skin friction coefficient. It is noted that both the homogeneous and heterogeneous reaction strengths reduced the concentration profile.

We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension $n$ with normal bundles isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus (n-1-p)}$ for some nonnegative integer $p$. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of $p=1$ and $n \leq 5$, which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role., Comment: To appear in Journal de Math\'ematiques Pures et Appliqu\'ees

In this paper, we present a method of computer calculation of Green functions in the form of generalized power series for third-order linear differential equations admitting regular singularities. For specific boundary-value problems, we construct Green functions by using the software proposed.

We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton–Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Lienard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Lienard equations, we use discrepancies in the original equations.

In this study, a new modification of the homotopy perturbation method (HPM) is introduced for various order boundary value problems (BVPs). In this modification, HPM is hybrid with least square optimizer and named as the least square homotopy perturbation method (LSHPM). The proposed scheme is tested against various linear and nonlinear BVPs (second to seventh order DEs). Validity of the obtained solutions is confirmed by finding absolute errors. To analyze the efficiency of the proposed scheme, tested problems have also been solved through HPM and results are compared with LSHPM. Furthermore, obtained results are also compared with other numerical schemes available in literature. Analysis reveals that LSHPM is a consistent and effective scheme which can be used for more complex BVPs in science and engineering.

We consider the nearly nonstationary autoregressive model yt = qnyt−1+ut, where qn = 1−c/n, c is a fixed constant, and {ut, t ≽ 1} is a sequence of innovations belonging to the domain of attraction of a stable distribution with index 0 < α < 2. We construct a composite quantile regression estimator for the autoregressive coefficient and establish the asymptotic distribution of this estimator under some mild conditions.

We study a nonlinear Bianchi equation which contains power nonlinearities in unknown function and its first derivatives. The method of separation of variables is used for investigation. An exponential solution is found in the case of an autonomous equation with linearly homogeneous right-hand side. It is shown that the equation has a solution in the form of a polylinear function in the case when the right-hand side of the equation contains the product of power functions of independent variables. Also, we have found solutions in the form of linear combination of exponents and in the form of generalized polynomials. The conditions on the parameters of the equation are given under which the above-mentioned solutions exist. Theorems determining the possibility of decreasing the dimension of the equation are proved. In particular, the initial equation is reduced to an ordinary differential equation for the solutions in the form of one- dimensional traveling waves, and it is reduced to a partial differential equation of lesser dimension for the solutions in the form of multi-dimensional traveling waves. In the latter case, a solution is found in the form of a generalized polynomial in linear combinations of independent variables.

In this article we consider certain linear time-varying control systems and investigate their stability using structured singular values ($\mu$-values). We use the low rank ordinary differential equations based methodology to compute the lower bounds for $\mu$-values. The inner-outer algorithm computes the local extremizer of an admissible perturbation and adjusts the desired perturbation level. Further, we present a comparison of our results via the well-known MATLAB routine mussv which is available in MATLAB control toolbox.

Let e > 0 be a small constant. We prove that whenever η is real and constants λi satisfy some necessary conditions, then there exist infinitely many prime triples p1, p2, p3 satisfying the inequality |λ1p1+λ2p2+λ3p3+η| < e and such that p3 = x2 + y2 + 1.

We investigate spectral properties of a nonself-adjoint problem for a differential operator of order 2n with nonlocal conditions that are perturbations of Sturm-type strongly regular self-adjoint conditions. We study the cases of problems with Birkhoff regular and irregular perturbations of boundary conditions. A system of eigenfunctions of the multipoint problem is constructed. We establish sufficient conditions under which this system is complete and, under certain additional assumptions, forms a Riesz basis.

We present sufficient conditions for fixed-time stability for a wide class of neural networks described by a system of differential equations with right-hand side satisfying the Carathéodory conditions. In contrast to the results given in the literature, where the settling-time function is estimated by an unknown Lyapunov function, we estimate the settling-time by a known function. In addition, the settling-time function does not depend on the initial values. We also give numerical examples, which confirm the theoretical results.

Population models of diabetes using ordinary differential equations are reviewed. They are refined by incorporating non-diabetics, prediabetics, low awareness prediabetics, awareness prediabetics, ...

We consider a nonlinear nonautonomous system of two ordinary differential equations with a stable fixed point and assume that the non-Hamiltonian part of the system tends to zero at infinity. We examine the asymptotic behavior of a two-parameter family of solutions that start from a neighborhood of the stable equilibrium. The proposed construction of asymptotic solutions is based on the averaging method and the transition in the original system to new dependent variables, one of which is the angle of the limit Hamiltonian system, and the other is the Lyapunov function for the complete system.

In this study, a new computing model is developed using the strength of feed-forward neural networks with the Levenberg–Marquardt scheme-based backpropagation technique (NN-BLMS). It is used to find a solution for the nonlinear system obtained from the governing equations of the magnetohydrodyanmic (MHD) boundary layer flow over a stretching sheet. Moreover, the partial differential equations (PDEs) for the MHD boundary layer flow over a stretching sheet are converting into ordinary differential equations (ODEs) with the help of similarity transformation. A dataset for the proposed NN-BLMM-based model is generated at different scenarios by a variation of various embedding parameters: Deborah number β and magnetic parameter (M). The training (TR), testing (TS), and validation (VD) of the NN-BLMS model are evaluated in the generated scenarios to compare the obtained results with the reference results. For the fluidic system convergence analysis, a number of metrics, such as the mean square error (MSE), error histogram (EH), and regression (RG) plots, are utilized for measuring the effectiveness and performance of the NN-BLMS infrastructure model. The experiments showed that comparisons between the results of proposed model and the reference results match in terms of convergence up to E-02 to E-10. This proves the validity of the NN-BLMS model. Furthermore, the results demonstrated that there is a decrease in the thickness of the boundary layer by increasing the Deborah number and magnetic parameter. The importance of the experiment can be seen due to its industrial applications such as MHD power generation, MHD generators, and MHD pumps.

In this paper, we propose a novel reduced differential transform method (RDTM) to compute analytical and semianalytical approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions. The performance of the proposed method was analyzed and compared with a convergent series solution form with easily computable coefficients. The behavior of approximated series solutions at different values of fractional order α and its modeling in 2-dimensional and 3-dimensional spaces are compared with exact solutions using MATLAB graphical method analysis. Moreover, the physical and geometrical interpretations of the computed graphs are given in detail within 2- and 3-dimensional spaces. Accordingly, the obtained approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions exactly fit with exact solutions. Hence, the proposed method reveals reliability, effectiveness, efficiency, and strengthening of computed mathematical results in order to easily solve fractional order Airy’s type differential equations.

In the analysis of the dynamics of the solutions of ordinary differential equations we can observe whether or not small variations or perturbations in the initial conditions produce small changes in the future; this intuitive idea of stability was formalized and studied by Lyapunov, who presented methods for the stable analysis of differential equations. For linear or nonlinear systems, we can also analyze the stability using criteria to obtain Hurwitz type polynomials, which provide conditions for the analysis of the dynamics of the system, studying the location of the roots of the associated characteristic polynomial. In this paper we present a stability study of a Lotka-Volterra type model which has been modified considering the carrying capacity or support in the prey and time delay in the predator, this stable analysis is performed using stability criteria to obtain Hurwitz-type polynomials.

The Darboux problem and the definition of the Riemann–Hadamard function are presented for the Bianchi equation of the fourth order—a linear fourth-order equation with four independent variables that has a dominant derivative not containing multiple differentiation with respect to any of the independent variables. Sufficient conditions on the coefficients of this equation under which its Riemann–Hadamard function allows explicit construction in terms of hypergeometric functions are obtained.

We state and study the first and second boundary value problems and the transmission problem with complicated boundary conditions containing singular functions for three-dimensional filtration flows in an inhomogeneous medium. The flow sources are arbitrary discrete sources and are located both on the boundaries and outside the boundaries. When boundaries are modeled by canonical surfaces (a plane and a sphere), the solutions of the problem can be presented in closed form. In the case of arbitrary closed smooth boundary surfaces with a point sink (source) located on them, the generalized double layer potential is used; this permits one to reduce the transmission problem and the second boundary value problem to singular and hypersingular integral equations, respectively. The problems studied here are mathematical models of three-dimensional filtration processes in heterogeneous media, which are of interest, for example, for the practice of developing natural oil-bearing (water-bearing) soil layers.

We derive volume integral equations with time delay describing nonstationary problems of acoustic field scattering by transparent three-dimensional structures. An efficient method for the numerical solution of the resulting equations is proposed.

For a fractional diffusion equation with reaction coefficient depending only on the first two components of the spatial variable $$x=(x_1,x_2,x_3)\in \mathbb {R}^3 $$ and on time $$t\geq 0 $$ , we consider the inverse problem of determining this coefficient under the assumption that the initial value at $$t=0 $$ is known for the solution of the equation and the boundary value at $$ x_3=0$$ is given as an additional condition. This inverse problem is reduced to equivalent integral equations, and we apply the contraction mapping principle to prove the existence of solutions of these equations. Local existence and global uniqueness theorems are proved. We also obtain a stability estimate for the solution of the inverse problem.

In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.

We study a linear system of integral equations of the first kind arising in the problem of wave diffraction by a local permeable body. The equivalence of the system and the original boundary value diffraction problem is established. The existence and uniqueness of the solution of the system and the dissipativity of its matrix operator are proved. The questions of the existence of nonradiating currents and their relation to the properties of the matrix operator of the system are considered.

The existence of uniform minimal trajectory and global attractors for weak solutions of the nonautonomous modified Kelvin–Voigt model is proved.

We use the Carleman function method to establish a criterion for the solvability of the Cauchy problem for the linearized Navier–Stokes equations in a multidimensional space and construct a regularized solution.

For the equation $$u_{{xx}}+yu_{{yy}}+\alpha u=0 $$ with a parameter $$\alpha \leq -1/2 $$ in the mixed domain lying in the right half-plane and bounded by the half-axis $$y\geq 0$$ and the characteristic $$ x=2\sqrt {-y}$$ , we consider the Neumann–Tricomi problem in which the values of the normal derivative of the unknown function are given on the half-axis, the values of the function itself are given on the characteristic, and matching conditions are specified on the singular line. The solution is sought in the class of functions having singularities at infinity of order no higher than a given one. Sufficient conditions on the input data under which the problem is uniquely solvable are obtained. The proof is carried out by the integral equation method.

Volterra integro-differential equations with integral operator kernels representable by Stieltjes integrals of the exponential function are studied. The approach is based on the study of one-parameter semigroups for linear evolution equations. A method for reducing the original initial value problem for a model integro-differential equation with operator coefficients in a Hilbert space to the Cauchy problem for a first-order differential equation in an extended function space is presented. The existence of a contraction $$C_0 $$ -semigroup is proved. As a corollary, we establish the well-posed solvability of the resulting Cauchy problem for the first-order differential equation in an extended function space and the initial value problem for the original abstract integro-differential equation and indicate a relation between their solutions.

We establish the uniqueness of the solution of the inverse problem for the Helmholtz equation in which the refraction coefficient, which describes the properties of a bounded inhomogeneity in a wave medium, is sought. The initial data for determining the desired coefficient are the absolute values of the solutions corresponding to either point sources concentrated on a segment or spherically symmetric distributed sources with centers on the segment. The measurements of the wave field are carried out in a plane domain that does not meet the inhomogeneity localization area.

Exact a priori estimates are obtained for the solution of an integral equation with sum kernel, a power-law nonlinearity, and an inhomogeneity in the linear part. With these estimates, we use the weighted metric method to prove a global theorem on the existence and uniqueness of a solution in the cone of nonnegative functions continuous on the positive half-line. It is shown that the solution can be found by the successive approximation method, and an estimate is found for the rate of convergence of these approximations to the exact solution. Examples are given to illustrate the results obtained.

In this study, inspired by the concept of B-metric-like space (BMLS), we introduce the concept of orthogonal B-metric-like space (OBMLS) via a hybrid pair of operators. Additionally, we establish the concept of orthogonal dynamic system (ODS) as a generalization of the dynamic system (DS), which improves the existing results for analysies such as those presented here. By applying this, some new refinements of the F⊥-Suzuki-type (F⊥-ST) fixed-point results are presented. These include some tangible instances, and applications in the field of nonlinear analysis are given to highlight the usability and validity of the theoretical results.

We investigate a case of the generalized Korteweg – De Vries Burgers equation. Our aim is to demonstrate the need for the application of further methods in addition to using Lie Symmetries. The solution is found through differential topological manifolds. We apply Lie’s theory to take the PDE to an ODE. However, this ODE is of third order and not easily solvable. It is through differentiable topological manifolds that we are able to arrive at a solution

In this study, a new computing model is developed using the strength of feedforward neural networks with the Levenberg–Marquardt method- (NN-BLMM-) based backpropagation technique. It is used to find a solution for the nonlinear system obtained from the governing equations of Falkner–Skan with heat transfer (FSE-HT). Moreover, the partial differential equations (PDEs) for the unsteady squeezing flow of heat and mass transfer of the viscous fluid are converted into ordinary differential equations (ODEs) with the help of similarity transformation. A dataset for the proposed NN-BLMM-based model is generated in different scenarios by a variation of various embedding parameters, Deborah number ( β ) and Prandtl number (Pr). The training (TR), testing (TS), and validation (VD) of the NN-BLMM model are evaluated in the generated scenarios to compare the obtained results with the reference results. For the fluidic system convergence analysis, a number of metrics such as the mean square error (MSE), error histogram (EH), and regression (RG) plots are utilized for measuring the effectiveness and performance of the NN-BLMM infrastructure model. The experiments showed that comparisons between the results of the proposed model and the reference results match in terms of convergence up to E-05 to E-10. This proves the validity of the NN-BLMM model. Furthermore, the results demonstrated that there is an increase in the velocity profile and a decrease in the thickness of the thermal boundary layer by increasing the Deborah number. Also, the thickness of the thermal boundary layer is decreased by increasing the Prandtl number.

In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.

This study is devoted to the investigation of nonlinear systems of fourth-order boundary value problems. Namely, using some techniques from matrix analysis and ordinary differential equations, a Lyapunov-type inequality providing a necessary condition for the existence of nonzero solutions is obtained. Next, an estimate involving generalized eigenvalues is derived as an application of our main result.

A regularizing algorithm is presented for solving the problem of dynamic reconstruction of an unknown input in a nonlinear vector differential equation. The algorithm is stable under information noise and computational errors.

Abstract We consider the spectral problem for a Dirac operator with arbitrary two-point boundary conditions and an arbitrary complex-valued integrable potential. The existence of nontrivial boundary value problems of this type with an unbounded growth of the multiplicity of eigenvalues is established.

We consider the Cauchy problem for the Emden–Fowler equation $$y^{\prime {}\prime }-x^ay^{\sigma }=0 $$ with parameters $$a\in \mathbb {R} $$ and $$\sigma

We study a system of first-order partial differential equations with generalized functions as coefficients that describes the heat and mass transfer process in domains with thin inclusions. It is proved that the operator of the problem is continuous and injective, and a theorem on the existence and uniqueness of a generalized solution is also established.

For the equation $$u_{{xx}}+yu_{{yy}}+\alpha u_y=0 $$ with parameter $$\alpha \leq -1/2 $$ given in the mixed domain that is the rectangle $$[0,1]\times [-\beta ,\gamma ]$$ , where $$\beta >0 $$ and $$\gamma >0 $$ , we investigate Dezin’s problem in which a periodicity condition is set on the vertical rectangle sides, the values of the unknown function are specified on the top side, matching conditions are given on the singular line, and a nonlocal condition relating the values of the unknown function on the singular line to the values of the normal derivative of this function is specified on the bottom side. The solution of the problem is constructed in the form of a series. Sufficient conditions ensuring the existence of a solution are found for the given functions and the parameters $$\beta $$ and $$\gamma $$ . A uniqueness criterion is established.

We study an initial–boundary value problem for an equation of the mixed parabolic-hyperbolic type in a rectangular parallelepiped. A uniqueness criterion is established. The solution is constructed in the form of a series in an orthogonal function system. The problem of small denominators depending on two positive integer arguments arises when justifying the convergence of this series. Sufficient conditions are found for the uniform separation of the denominators from zero; this permits us to prove the convergence of the series in the class of regular solutions and the stability of the solution with respect to perturbations of the boundary functions.

Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomena in hydrodinamics and other ones. Notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existence and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations. In the present paper, a third order nonlinear partial equation containing a $$p$$ -Laplacian is studied. Six classes of its exact solutions are built. They are expressed in terms of elementary functions, quadratures and special functions (solutions of some ordinary differential equations).

For a linear inhomogeneous system of first-order partial differential equations with three independent variables, we prove the existence and uniqueness of a solution of the Darboux problem, which is constructed in terms of the Riemann–Hadamard matrix defined in the paper.