F. D. Zaman

A note on a symmetry analysis and exact solutions of a nonlinear fin equation

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and solutions not appearing elsewhere.

Initial inverse problem in heat equation with Bessel operator

Abstract We investigate the inverse problem involving recovery of initial temperature from the information of final temperature profile in a disc. This inverse problem arises when experimental measurements are taken at any given time, and it is desired to calculate the initial profile. We consider the usual heat equation and the hyperbolic heat equation with Bessel operator. An integral representation for the problem is found, from which a formula for initial temperature is derived using Picards criterion and the singular system of the associated operators.

A fractional diffusion equation model for cancer tumor

In this article, we consider cancer tumor models and investigate the need for fractional order derivative as compared to the classical first order derivative in time. Three different cases of the net killing rate are taken into account including the case where net killing rate of the cancer cells is dependent on the concentration of the cells. At first, we use a relatively new analytical technique called q-Homotopy Analysis Method on the resulting time-fractional partial differential equations to obtain analytical solution in form of convergent series with easily computable components. Our numerical analysis enables us to give some recommendations on the appropriate order (fractional) of derivative in time to be used in modeling cancer tumor.

Symmetries and integrability of a fourth-order Euler–Bernoulli beam equation

The complete symmetry group classification of the fourth-order Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solut...

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.

Wave equation on spherically symmetric Lorentzian metrics

Wave equation on a general spherically symmetric spacetime metric is constructed. Noether symmetries of the equation in terms of explicit functions of θ and ϕ are derived subject to certain differential constraints. By restricting the metric to flat Friedman case the Noether symmetries of the wave equation are presented. Invertible transformations are constructed from a specific subalgebra of these Noether symmetries to convert the wave equation with variable coefficients to the one with constant coefficients.

Invariant boundary value problems for a fourth-order dynamic Euler-Bernoulli beam equation

We obtain the complete Lie symmetry group classification of the dynamic fourth-order Euler-Bernoulli partial differential equation, where the elastic modulus, the area moment of inertia are constants and the applied load is a nonlinear function. In the Lie analysis, the principal Lie algebra which is two-dimensional extends in three cases, viz., the linear, the exponential, and the general power law. For each of the nontrivial cases, we determine symmetry reductions to ordinary differential equations which are of order four. In only one case related to the power law we are able to have a compatible initial-boundary value problem for a clamped end and a free beam. For these cases we deduce the corresponding fourth-order ordinary differential equations with appropriate boundary conditions. We provide an asymptotic solution for the reduced fourth-order ordinary differential equation corresponding to a clamped or free beam.

Investigation of the Initial Inverse Problem in the Heat Equation

We investigate the inverse problem in the heat equation involving the recovery of the initial temperature front measurements of the final temperature. This problem is extremely ill-posed and it is believed that only information in the first few modes can be recovered by classical methods. We will consider this problem with a regularizing parameter which approximates and regularizes the heat conduction model

Cooling of a plate with general boundary conditions

We consider steady state temperature distribution in a homogeneous rectangular infinite plate the lower part of which is cooled by a fluid flowing at a constant velocity while the upper part satisfies the general mixed boundary conditions. The Wiener-Hopf method has been used to obtain the solution in the infinite series form and some special cases have been discussed.

Acoustic waves in a layered inhomogeneous ocean

Abstract We consider a layered ocean of finite depth in which the lower layer is assumed to have depth dependent properties. The seabed is considered to be either rigid or of a reflecting type. The perturbation method is used to obtain the eigenvalues and the eigenfunctions of the depth equation in the case of both the rigid and reflecting seabed. The corrections to the eigenvalues and eigenfunctions are numerically computed from the perturbation formulae in some case of interest.