Andrei Polyanin

Partial differential equation

In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. They find their generalization in stochastic partial differential equations. Just as ordinary differential equations often model dynamical systems, partial differential equations often model multidimensional systems.

Handbook of first order partial differential equations

Linear Equations with Two Independent Variables. Linear Equations with Three or More Independent Variables. Nonlinear Equations. Equations with Two Independent Variables. Quadratic in Derivatives. Nonlinear Equations with Two Independent Variables or General Form Nonlinear Equations with Three or More Independent Variables. Supplement Solution of Differential Equations Through the CONVODE Software.

A Concise Handbook of Mathematics, Physics, and Engineering Sciences

MATHEMATICS Arithmetic and Elementary Algebra Elementary Functions Elementary Geometry Analytic Geometry Algebra Limits and Derivatives Integrals Series Functions of Complex Variables Integral Transforms Ordinary Differential Equations Partial Differential Equations Special Functions and Their Properties Probability Theory PHYSICS Physical Foundations of Mechanics Molecular Physics and Thermodynamics Electrodynamics Oscillations and Waves Optics Quantum Mechanics. Atomic Physics Quantum Theory of Crystals Elements of Nuclear Physics ELEMENTS OF APPLIED AND ENGINEERING SCIENCES Dimensions and Similarity Mechanics of Point Particles and Rigid Bodies Elements of Strength of Materials Hydrodynamics Mass and Heat Transfer Electrical Engineering Empirical and Engineering Formulas and Criteria for Their Applicability SUPPLEMENTS Integrals Integral Transforms Orthogonal Curvilinear Systems of Coordinates Ordinary Differential Equations Some Useful Electronic Mathematical Resources Index References appear at the end of each chapter.

Generalized Separation of Variables in Nonlinear Heat and Mass Transfer Equations

Abstract We outline generalized separation of variables as applied to nonlinear second-order partial differential equations (PDEs). In this context, we suggest a method for constructing exact solutions of nonlinear PDEs. The method involves searching for transformations that “reduce the dimensionality” of the equation. New families of exact solutions of 3D nonlinear elliptic and parabolic equations that govern processes of heat and mass transfer in inhomogeneous anisotropic media are described. Moreover, the method makes it possible to construct exact solutions of nonlinear wave equations. We also present solutions for three families of equations with logarithmic heat sources; the solutions are obtained by nonlinear separation of variables.

Nonlinear delay reaction–diffusion equations with varying transfer coefficients: Exact methods and new solutions

Abstract The paper deals with one-dimensional nonlinear delay reaction–diffusion equations with varying transfer coefficients of the form u t = [ G ( u ) u x ] x + F ( u , u ) , where u = u ( x , t ) and u = u ( x , t − τ ) , with τ denoting the delay time. Generalized and functional separable solutions for this class of equations have been obtained and presented for the first time; these equations have not been known to have such solutions so far. To construct these solutions and solutions of some other delay PDEs, we developed a few exact methods that rely on using invariant subspaces for corresponding nonlinear differential operators. Many of the results are extendable to more complex nonlinear reaction–diffusion equations with several delay times, τ 1 , … , τ m , and equations with time-varying delay, τ = τ ( t ) . All of the equations considered involve several free parameters (or an arbitrary function) and so their solutions can be suitable for testing approximate analytical and numerical methods for nonlinear delay reaction–diffusion equations. The exact methods described may also be applied to other classes of nonlinear delay PDEs.

Parametrically defined nonlinear differential equations and their solutions: Applications in fluid dynamics

The study deals with parametrically defined ordinary differential equations, practically unaddressed in the literature. It finds the general solutions for three classes of first- and second-order nonlinear ODEs of this kind. The solutions are further used to construct new exact solutions to the equations of an unsteady axisymmetric boundary layer with pressure gradient on a body of revolution of arbitrary shape. Also the paper suggests a short list of essential problems for nonlinear ODEs and PDEs defined parametrically that need to be addressed in the future.

Parametrically defined nonlinear differential equations, differential–algebraic equations, and implicit ODEs: Transformations, general solutions, and integration methods

The study deals with nonlinear ordinary differential equations defined parametrically by two relations; these arise in fluid dynamics and are a special class of coupled differential–algebraic equations. We propose a few techniques for reducing such equations, first or second order, to systems of standard ordinary differential equations as well as techniques for the exact integration of these systems. Several examples show how to construct general solutions to some classes of nonlinear equations involving arbitrary functions. We specify a procedure for the numerical solution of the Cauchy problem for parametrically defined differential equations and related differential–algebraic equations. The proposed techniques are also effective for the numerical integration of problems for implicitly defined equations.

Nonlinear delay reaction–diffusion equations: Traveling-wave solutions in elementary functions

Abstract The paper deals with nonlinear delay reaction–diffusion equations of the form u t = a u x x + F ( u , u ) , where u = u ( x , t ) and u = u ( x , t − τ ) , with τ denoting the delay time. We present a number of traveling-wave solutions of the form u = w ( z ) , z = k x + λ t , that can be represented in terms of elementary functions. We consider equations with quadratic, power-law, exponential and logarithmic nonlinearities as well as more complex equations with the kinetic function dependent on one to four arbitrary functions of a single argument. All of the solutions obtained involve free parameters and so may be suitable for solving certain model problems as well as testing numerical and approximate analytical methods for delay reaction–diffusion equations and more complex nonlinear delay PDEs.

Non-monotonic blow-up problems: Test problems with solutions in elementary functions, numerical integration based on non-local transformations

Abstract We consider blow-up problems having non-monotonic singular solutions that tend to infinity at a previously unknown point. For second-, third-, and fourth-order nonlinear ordinary differential equations, the corresponding multi-parameter test problems allowing exact solutions in elementary functions are proposed for the first time. A method of non-local transformations, that allows to numerically integrate non-monotonic blow-up problems, is described. A comparison of exact and numerical solutions showed the high efficiency of this method. It is important to note that the method of non-local transformations can be useful for numerical integration of other problems with large solution gradients (for example, in problems with solutions of boundary-layer type).

Exact solutions and qualitative features of nonlinear hyperbolic reaction—diffusion equations with delay

New classes of exact solutions to nonlinear hyperbolic reaction—diffusion equations with delay are described. All of the equations under consideration depend on one or two arbitrary functions of one argument, and the derived solutions contain free parameters (in certain cases, there can be any number of these parameters). The following solutions are found: periodic solutions with respect to time and space variable, solutions that describe the nonlinear interaction between a standing wave and a traveling wave, and certain other solutions. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Conditions for the global instability of solutions to a number of reaction—diffusion systems with delay are derived. The generalized Stokes problem subject to the periodic boundary condition, which is described by a linear diffusion equation with delay, is solved.