# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Continuous numerical solutions and error bounds for time dependent systems of partial differential equations: Mixed problems. (English) Zbl 0831.65102
For the mixed problems described by the equation $u_t(x,t) - D(t) u_{xx} (x,t) = 0$, $0 < x < p$, $t > 0$ in the bounded domain $\Omega$ $(t_0, t_1) = [0, p] \times [t_0, t_1]$, subject to boundary conditions $u(x,0) = F(x)$ and initial conditions $u(0,t) = u(p,t) = 0$ the authors construct continuous numerical solutions with prefixed accuracy. They assume that $u(x,t)$ and $F(x)$ are $r$-component vectors and $D(t)$ is a $\bbfC^{r \times r}$ valued two-times continuously differentiable function, so that $D(t_1) D(t_2) = D(t_2) D(t_1)$ for $t_2 \geq t_1 > 0$ and there exists a positive number $\delta$ such that every eigenvalue $z$ of the form $(D(t) + D^H(t))/2$ with $t > 0$ is bigger than $\delta$. Such coupled partial differential equations appear in many different problems, for example in magnetohydrodynamic flows, in the study of temperature distribution within a composite heat conductor, mechanics, diffusion problems, nerve conduction problems, biochemistry, armament models etc.

##### MSC:
 65M70 Spectral, collocation and related methods (IVP of PDE) 65M15 Error bounds (IVP of PDE) 35K15 Second order parabolic equations, initial value problems
Full Text:
##### References:
 [1] Sezgin, M.: Magnetohydrodynamic flow in a rectangular duct. Int. J. Numer. methods fluids 7, 697-718 (1987) · Zbl 0639.76112 [2] Cannon, J. R.; Klein, R. E.: On the observability and stability of the temperature distribution in a composite heat conductor. SIAM J. Appl. math. 24, 596-602 (1973) · Zbl 0236.35026 [3] Paul, B.: Analytical dynamics of mechanisms--A computer oriented overview. Mech. Mach theory 10, 481-507 (1975) [4] Whittaker, E. T.: A treatise of the analytic dynamics of particles and rigid bodies. (1974) [5] Lee, A. I.; Hill, J. H.: On the general linear coupled systems for diffusion in media with two diffusivities. J. math. Anal. appl. 89, 530-538 (1992) [6] Morimoto, H.: Stability in the wave equation coupled with heat flow. Numerische math. 4, 136-145 (1962) · Zbl 0235.65076 [7] Hodgkin, A. L.; Huxley, A. F.: A quantitative description of membrane current and its applications in the giant axon of loligo. J. physiol. 117, 500-544 (1952) [8] Mascagni, M.: An initial-boundary value problem of physiological significance for equations of nerve conduction. Comm. pure and appl. Math. 42, 213-227 (1989) · Zbl 0664.92007 [9] King, A. I.; Chou, C. C.: Mathematical modelling, simulation and experimental testing of biochemical systems crash response. J. biomech. 9, 301-317 (1976) [10] Golpasamy, K.: An arms race with deteriorating armaments. Math. biosci. 37, 191-203 (1977) · Zbl 0368.90101 [11] Ames, W. F.: Numerical methods for partial differential equations. (1977) · Zbl 0577.65077 [12] Ciarlet, P. G.; Lions, J. L.: 2nd edition handbook of numerical analysis. Handbook of numerical analysis (1990) [13] Vermuri, V.; Karples, W. J.: Digital computer treatment of partial differential equations. (1981) [14] Freedman, H. I.: Functionally commutative matrices and matrices with constant eigenvectors. Linear and multilinear algebra 4, 107-113 (1976) [15] Freedman, H. I.; Lawson, J. D.: Systems with constant eigenvectors with applications to exact and numerical solutions of ordinary differential equations. Linear algebra and its applications 8, 369-374 (1974) · Zbl 0306.65018 [16] Mikhailov, M. D.; Özişik, M. N.: Unified analysis and solutions of heat and mass diffusion. (1984) [17] Nicolis, G.; Prigogine, I.: Self-organization in nonequilibrium systems. (1977) · Zbl 0363.93005 [18] Özişik, M. N.: Boundary value problems of heat conduction. (1968) [19] Jódar, L.: Computing accurate solutions for coupled systems of second order partial differential equations II. Int. J. Computer math. 46, 63-76 (1992) · Zbl 0813.65114 [20] Jódar, L.; Ponsoda, E.: Analytic approximate solutions and error bounds for linear matrix differential equations appearing in control. Control and cybernetics 21, No. 3/4, 21-34 (1992) · Zbl 0781.34015 [21] Golub, G. H.; Van Loan, C. F.: Matrix computations. (1989) · Zbl 0733.65016 [22] Ortega, J. M.; Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. (1970) · Zbl 0241.65046 [23] Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. of the royal inst. Of techn. (1959) · Zbl 0085.33401 [24] Flett, T. M.: Differential analysis. (1980) · Zbl 0442.34002 [25] Hairer, E.; Nørselt, S. P.; Wanner, G.: Solving ordinary differential equations I. (1980) [26] Churchill, R. V.; Brown, J. W.: Fourier series and boundary value problems. (1978) · Zbl 0378.42001 [27] Stoer, J.; Bulirsch, R.: Introduction to numerical analysis. (1980) · Zbl 0423.65002 [28] Apostol, T. M.: Mathematical analysis. (1977) [29] Zygmund, A.: 2nd edition trigonometric series. Trigonometric series (1977) [30] Moler, C. B.; Vanloan, C. F.: Nineteen dubious ways to compute the exponential of a matrix. SIAM review 20, 801-836 (1978) · Zbl 0395.65012