Comparison of differential representations for radially symmetric Stokes flow.
Computation of radial solutions of semilinear equations.
Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain
We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations (SE) f(x,u,Dx u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point and . The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 . The criterion of convergence of a formal solution of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal...
Convergence of iterative methods applied to generalized Fisher equation.
Convergence of power series along vector fields and their commutators; a Cartan-Kähler type theorem
We study convergence of formal power series along families of formal or analytic vector fields. One of our results says that if a formal power series converges along a family of vector fields, then it also converges along their commutators. Using this theorem and a result of T. Morimoto, we prove analyticity of formal solutions for a class of nonlinear singular PDEs. In the proofs we use results from control theory.
Convergence, via summability, of formal power series solutions to a certain class of completely integrable Pfaffian systems
Cooling of a plate with general boundary conditions.
Eine verallgemeinerte Darboux-Gleichung I
Einige Eigenschaften der Lösung der Wellengleichung mit der Korrektion von Rayleigh
Exact, approximate solutions and error bounds for coupled implicit systems of partial differential equations.
Expansions of solutions of higher order evolution equations in series of generalized heat polynomials.
Explicit solution for Lamé and other PDE systems
We provide a general series form solution for second-order linear PDE system with constant coefficients and prove a convergence theorem. The equations of three dimensional elastic equilibrium are solved as an example. Another convergence theorem is proved for this particular system. We also consider a possibility to represent solutions in a finite form as partial sums of the series with terms depending on several complex variables.
Explicit solutions of generalized nonlinear Boussinesq equations.
Explicit Solutions of Nonlocal Boundary Value Problems, Containing Bitsadze-Samarskii Constraints
MSC 2010: 44A35, 35L20, 35J05, 35J25In this paper are found explicit solutions of four nonlocal boundary value problems for Laplace, heat and wave equations, with Bitsadze-Samarskii constraints based on non-classical one-dimensional convolutions. In fact, each explicit solution may be considered as a way for effective summation of a solution in the form of nonharmonic Fourier sine-expansion. Each explicit solution, may be used for numerical calculation of the solutions too.
Exponential-Bessel partial differential equation and Fox’s -function
Exponential-type Nagumo norms and summability of formal solutions of singular partial differential equations
In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at . Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the -summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.
Extension of formal power series solutions
External ellipsoidal harmonics for the Dunkl-Laplacian.
Formal and Convergent Solutions of Singular Partial Differential Equations.
Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularity
In this paper, we calculate the formal Gevrey index of the formal solution of a class of nonlinear first order totally characteristic type partial differential equations with irregular singularity in the space variable. We also prove that our index is the best possible one in a generic case.