Mixed problem for a first-order partial differential equation with involution and periodic boundary conditions

The Fourier method is used to find a classical solution of the mixed problem for a first-order differential equation with involution and periodic boundary conditions. The application of the Fourier method is substantiated using refined asymptotic formulas obtained for the eigenvalues and eigenfunctions of the corresponding spectral problem. The Fourier series representing the formal solution is transformed using certain techniques, and the possibility of its term-by-term differentiation is proved. Minimal requirements are imposed on the initial data of the problem.     

A nonlocal problem for a first-order partial differential equation

In this paper we study a nonlocal problem for a first-order partial differential equation with an integral condition instead of the standard boundary one. We prove that the problem under consideration is uniquely solvable.     

Boundary value problem for a first-order partial differential equation with a fractional discretely distributed differentiation operator

We solve a boundary value problem for a first-order partial differential equation in a rectangular domain with a fractional discretely distributed differentiation operator. The fractional differentiation is given by Dzhrbashyan–Nersesyan operators. We construct a representation of the solution and prove existence and uniqueness theorems. The results remain valid for the corresponding equations with Riemann–Liouville and Caputo derivatives. In terms of parameters defining the fractional differential operator, we derive necessary and sufficient conditions for the solvability of the problem.     

An efficient method for solving nonlocal initial-boundary value problems for linear and nonlinear first-order hyperbolic partial differential equations

In this paper, we present a new approach to solve nonlocal initial-boundary value problems of linear and nonlinear hyperbolic partial differential equations of first-order subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problems into local initial-boundary value problems. Then we apply a modified Adomian decomposition method, which permits convenient resolution of these problems. Moreover, we prove this decomposition scheme applied to such nonlocal problems is convergent in a suitable Hilbert space, and then extend our discussion to include systems of first-order linear equations and other related nonlocal initial-boundary value problems.     

Solvability of the initial-boundary value problem for an integrodifferential equation

Under study is the well-posedness of the Cauchy problem for the nonstationary radiation transfer equation with generalized matching conditions at the interface between the media. We prove the existence of a unique strongly continuous semigroup of resolvents, estimate its order of growth, and consider the question of stabilization of the nonstationary solution.     

Boundary value problem for a multidimensional fractional partial differential equation

We construct a fundamental solution of a linear fractional partial differential equation. For an equation with Dzhrbashyan-Nersesyan fractional differentiation operators, we solve a boundary value problem and find a closed-form representation for its solution. The corresponding results for equations with Riemann-Liouville and Caputo derivatives are special cases of the assertions proved here.     

A method of characteristics solving the initial-boundary value problem of a hyperbolic differential equation of second order

Summary The ALGOL-procedure¹ char2 presented in this paper can be applied to the initial or initial-boundary value problem of a quasilinear hyperbolic differential equation of second order. A method of characteristics is combined with extrapolation to the limit. Thus, the results are very accurate. The same accuracy can also be obtained if the initial values are only piecewise smooth.Editor's Note: In this fascile, prepublication of algorithms from the Approximation series of the Handbook for Automatic Computation is continued. Algorithms are published in ALGOL 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones     

On the initial value problem for a certain partial differential equation

It is shown that the initial problem of the equationu _xx +u _yy =(tu _t ) _t has a unique solution satisfying a maximum principle. Moreover, a numerical scheme for its solution is proposed.     

Direct numerical method for an inverse problem of a parabolic partial differential equation

A coefficient inverse problem of the one-dimensional parabolic equation is solved by a high-order compact finite difference method in this paper. The problem of recovering a time-dependent coefficient in a parabolic partial differential equation has attracted considerable attention recently. While many theoretical results regarding the existence and uniqueness of the solution are obtained, the development of efficient and accurate numerical methods is still far from satisfactory. In this paper a fourth-order efficient numerical method is proposed to calculate the function u(x,t) and the unknown coefficient a(t) in a parabolic partial differential equation. Several numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.     

Inverse initial-boundary value problem for a fractional diffusion-wave equation with a non-carleman shift

An inverse initial-boundary value problem is considered for a linear inhomogeneous second-order equation with a fractional time derivative and with delay in the spatial coordinate.     

An initial-boundary value problem for a Sobolev-type strongly nonlinear dissipative equation

An initial-boundary value problem is considered for a model equation governing waves in crystalline semiconductors with allowance for strong spatial dispersion, linear dissipation, and sources of free charges. The weak generalized local-in-time solvability of the problem is proved. Sufficient conditions are obtained for the blowup of the solution and for global-in-time solvability. Two-sided estimates for the blowup time are derived.     

ON AN INVERSE INITIAL-BOUNDARY VALUE PROBLEM FOR A ONE-DIMENSIONAL HYPERBOLIC SYSTEM OF DIFFERENTIAL EQUATIONS

1.IntroductionTheproblemstodeterminethecoefficientsindifferentialequationsfromknownfunctionaloftheirsolutionsareoftencalledinverseproblems.Amongtheseproblemsthesimplestisaboutone-dimensionalwaveequations.Thisproblemcanbediscussedinthetimedomainorinthefreq…     

An initial-boundary value problem with mixed lateral conditions for heat equation

Sunto Viene studiato un problema con condizioni laterali miste per l'equazione del calore, nel caso in cui la ? superficie di separazione ? abbia un punto caratteristico. Viene dato un teorema di esistenza e di unicità in spazi con peso a regolarità variabile.

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Entrata in Redazione l'8 marzo 1978.

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Partially supported by Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Bologna, Italy.     

An iterative method for solving an inverse problem for a first-order nonlinear partial differential equation with estimates of guaranteed accuracy and the number of steps

For a partial differential equation simulating population dynamics, the inverse problem of determining its nonlinear right-hand side from an additional boundary condition is studied. This inverse problem is reduced to a functional equation, for which the existence and uniqueness of a solution is proven. An iterative method for solving this inverse problem is proposed. The accuracy of the method is estimated, and restrictions on the number of steps are obtained.     

The solution of the Cauchy problem of constructing an integral hypersurface of a first-order partial differential equation

A geometric interpretation is found for the solution of the Cauchy problem for a nonlinear first-order partial differential equation. The interpretation is based on the connection between Monge equations and nonlinear partial differential equations.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 123–126.     

Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green’s function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.