M. T. Mustafa

Applications of harmonic morphisms to gravity

We introduce the notion of gravity coupled to a horizontally conformal submersion as a modification of the well-known concept of gravity coupled to a harmonic map, thus obtaining a coupled gravity system with a more geometric flavor. By using integral techniques we determine the necessary conditions for coupling and cosmological constants. Finally, in the context of higher dimensional gravitation theory, we show that harmonic morphisms provide a natural ansatz to trigger spontaneous splitting and reduction of the gravity system coupled to a harmonic map on (4+D) (D⩾1) dimensional space–times.

Adomian Decomposition Method for a Nonlinear Heat Equation with Temperature Dependent Thermal Properties

The solutions of nonlinear heat equation with temperature dependent diffusivity are investigated using the modified Adomian decomposition method. Analysis of the method and examples are given to show that the Adomian series solution gives an excellent approximation to the exact solution. This accuracy can be increased by increasing the number of terms in the series expansion. The Adomian solutions are presented in some situations of interest.

A Bochner Technique for Harmonic Morphisms

We establish aWeitzenb ock formula for harmonicmor phisms between Riemannian manifolds and show that under suit able curvature conditions such a map is totally geodesic As an ap plication of the Weitzenb ock formula we obtain some non existence results of a global nature for harmonic morphisms and totally ge odesic horizontally conformal maps between compact Riemannian manifolds In particular it is shown that the only harmonic mor phisms from a Riemannian symmetric space of compact type to a compact Riemann surface of genus are the constant maps Introduction A smooth map M N between Riemannian manifolds is called a harmonic morphism if it preserves germs of harmonic functions i e if f is a real valued harmonic function on an open set V N then the com position f is harmonic on V M Due to a characterization obtained by B Fuglede and T Ishihara harmonic morphisms are precisely the harmonic maps which are horizontally weakly con formal The purpose of this paper is to develop a Bochner technique for harmonic morphisms extending the work of Eells Sampson to Mathematics Subject Classi cation E C

Symmetry analysis of wave equation on static spherically symmetric spacetimes with higher symmetries

Symmetry analysis of wave equation on all static spherically symmetric spacetimes admitting maximal isometry groups G10 or G7 or G6 is carried out. Symmetry algebras of the wave equation are found and their structural information-in the sense of Iwasawa decomposition-is obtained. Joint invariants of appropriate subalgebras are utilized to obtain many exact solutions of the wave equation on static spherically symmetric spacetimes.

Polynomial solutions of differential equations

AbstractA new approach for investigating polynomial solutions of differential equations is proposed. It is based on elementary linear algebra. Any differential operator of the form L(y)=∑k=0k=Nak(x)y(k), where ak is a polynomial of degree ≤ k, over an infinite field F has all eigenvalues in F in the space of polynomials of degree at most n, for all n. If these eigenvalues are distinct, then there is a unique monic polynomial of degree n which is an eigenfunction of the operator L, for every non-negative integer n. Specializing to the real field, the potential of the method is illustrated by recovering Bochners classification of second order ODEs with polynomial coefficients and polynomial solutions, as well as cases missed by him - namely that of Romanovski polynomials, which are of recent interest in theoretical physics, and some Jacobi type polynomials. An important feature of this approach is the simplicity with which the eigenfunctions and their orthogonality and norms can be determined, resulting in significant reduction in computational complexity of such problems. 2000 MSC: 33C45; 34A05; 34A30; 34B24.

A non-existence result for compact Einstein warped products

Warped products provide a rich class of physically significant geometric objects. The existence of compact Einstein warped products was questioned in Besse (1987 Einstein Manifolds, section 9.103). It is shown that there exists a metric on every compact manifold B such that (non-trivial) Einstein warped products, with base B, cannot be constructed.

Analytic solutions of initial-boundary-value problems of transient conduction using symmetries

Lie symmetry method is applied to find analytic solutions of initial-boundary-value problems of transient conduction in semi-infinite solid with constant surface temperature or constant heat flux condition. The solutions are obtained in a manner highlighting the systematic procedure of extending the symmetry method for a PDE to investigate BVPs of the PDE. A comparative analysis of numerical and closed form solutions is carried out for a physical problem of heat conduction in a semi-infinite solid bar made of AISI 304 stainless steel.

Restrictions on harmonic morphisms

We consider horizontally (weakly) conformal maps φ between Riemannian manifolds and calculate a formula for the Laplacian of the dilation of φ, using the language of moving frames. Applying this formula to harmonic horizontally (weakly) conformal maps or equivalently to harmonic morphisms we obtain a Weitzenbock formula similar to [?], and hence vanishing results for harmonic morphisms from compact manifolds of positive curvature. Further, a method is developed to obtain restrictions on harmonic morphisms from some non-compact and non-positively curved domains. Finally, a discussion of restrictions on harmonic morphisms between simply connected space forms is given.

THE STRUCTURE OF HARMONIC MORPHISMS WITH TOTALLY GEODESIC FIBRES

The structure of local and global harmonic morphisms between Riemannian manifolds, with totally fibres, is investigated. It is shown that non-positive curvature of the domain obstructs the existence of global harmonic morphisms with totally geodesic fibres and the only such maps from compact Riemannian manifolds of non-positive curvature are, up to a homothety, totally geodesic Riemannian submersions. Similar results are obtained for local harmonic morphisms with totally geodesic fibres from open subsets of non-negatively curved compact and non-compact manifolds. During the course, we prove non-existence of submersive harmonic morphisms with totally geodesic fibres from some important domains, for instance from compact locally symmetric spaces of non-compact type and open subsets of symmetric spaces of compact type.

A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations

Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.