a diffusion wave equation with two fractional derivatives of different order we analyse a diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains thus our model represents a generalized telegraph equation solutions to signalling and cauchy problems in terms of a series and integral representation are given classical wave and heat conduction equations are obtained as limiting cases

boundary element methods for optimization of distributed parameter systems a numerical method of solution is proposed for optimization problems of distributed parameter systems two model problems from continuum mechanics are investigated by means of constructing the problems as the steady state optimal control problems governed by elliptic partial differential equations the basis of the suggested method of solution lies in the space discretization of the necessary conditions for optimality by the boundary element method and the minimization of the performance indices by the conjugate gradient method of optimization

backward differentiation type formulas for volterra integral equations of the second kind numerical integration formulas are discussed which are obtained by differentiation of the volterra integral equation and by applying backward differentiation formulas to the resulting integro differential equation in particular the stability of the method is investigated for a class of convolution kernels the accuracy and stability behaviour of the method proposed in this paper is compared with that of i a block implicit runge kutta scheme and ii the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas the present method is particularly advantageous in the case of stiff volterra integral equations

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

boundedness of solutions of difference equations and application to numerical solution of volterra integral equations of the second kind abstract we study the difference equations obtained when some numerical methods for volterra integral equations of the second kind are applied to the linear test problem y t 1 0 t t vs y s ds t 0 with fixed stepsize h the resulting difference equations are of poincare type and we formulate a criterion for boundedness of solutions of these equations if the associated characteristic polynomial is a simple von neumann polynomial this result is then used in stability analysis of reducible quadrature methods for volterra integral equations

space time fractional zener wave equation the space time fractional zener wave equation describing viscoelastic materials obeying the time fractional zener model and the space fractional strain measure is derived and analysed this model includes waves with finite speed as well as non propagating disturbances the existence and the uniqueness of the solution to the generalized cauchy problem are proved special cases are investigated and numerical examples are presented

stability analysis and aerodynamic design optimization of euler equations using variational methods the one and two dimensional inviscid euler equations are formulated in an integro differential form for shape design sensitivity analysis and optimization the principal tool employed to derive the performance derivative sensitivity equations is the variational method which is a continuous alternative to the discrete sensitivity analysis along with the sensitivity equations the co state equations and their boundary conditions are derived thereafter based on the modal analysis of von neumann theory the stability limits of the co state equations are investigated the stability characteristics of the co state equations are compared with those of the euler equations finally using the criteria from the stability analysis of the state and co state equations some results of shape design optimization and sensitivity analysis for both quasi one and two dimensional euler equations are presented copyright 1999 john wiley sons ltd

a stability analysis of the trapezoidal method for volterra integral equations with completely positive kernels abstract the solution of the volterra integral equation with completely positive kernel y t 0t b t s y s ds u0 0t b t s g s ds t 0 is nonnegative and nonincreasing provided that g is nonincreasing and 0 g t u0 for any t 0 we prove that under some additional hypotheses this property is inherited by the solution of the recurrence relation resulting from applying the trapezoidal method to this equation

time distributed order diffusion wave equation ii applications of laplace and fourier transformations a cauchy problem for a time distributed order multi dimensional diffusion wave equation containing a forcing term is reinterpreted in the space of tempered distributions and a distributional diffusion wave equation is obtained the distributional equation is solved in the general case of weight function or distribution solutions are given in terms of solution kernels green s functions which are studied separately for two cases the first case is when the order of the fractional derivative is in the interval 0 1 while in the second case the order of the fractional derivative is in the interval 0 2 solutions of fractional diffusionwave and fractional telegraph equations are obtained as special cases numerical experiments are also performed an analogue of the maximum principle is also presented

the importance of diagonal dominance in the iterative solution of equations generated from the boundary element method the unsymmetric matrix equations generated from the boundary element method bem can be solved iteratively with convergence to the correct solution guaranteed if the boundary element system of equations can be first transformed into an equivalent diagonally dominant system a transformation is presented which selectively annihilates terms in the coefficient matrix of the system ax b until an equivalent row diagonally dominant system if available is obtained the new row diagonally dominant system is well suited for use with jacobi and gauss seidel point iterative equation solvers the diagonal dominizing transformation presented in this work is not suitable for large systems of equations but is useful as a research tool for studying the importance of diagonal dominance in the iterative solution of equations generated from the bem a simple laplacian problem is used to examine the structure of the bem equations and to introduce the diagonal dominizing transformation the importance of diagonal dominance is shown by comparing iterative convergence of positive definite symmetric positive definite and diagonally dominant systems of bem equations obtained from a plane strain elasticity problem

distributed order calculus and equations of ultraslow diffusion abstract we consider equations of the form d u t x u t x f t x t 0 x r n where d is a distributed order derivative that is d t 0 1 d t d d is the caputo dzhrbashyan fractional derivative of order is a positive weight function the above equation is used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement in this work we develop a mathematical theory of such equations study the derivatives and integrals of distributed order

time distributed order diffusion wave equation i volterra type equation a single order time fractional diffusion wave equation is generalized by introducing a time distributed order fractional derivative and forcing term while a laplacian is replaced by a general linear multi dimensional spatial differential operator the obtained equation is in the case of the laplacian called a time distributed order diffusion wave equation we analyse a cauchy problem for such an equation by means of the theory of an abstract volterra equation the weight distribution occurring in the distributed order fractional derivative is specified as the sum of the dirac distributions and the existence and uniqueness of solutions to the cauchy problem and the corresponding volterra type equation were proven for a general linear spatial differential operator as well as in the special case when the operator is laplacian

multi dimensional solutions of space time fractional diffusion equations green9s functions and propagators for the multi dimensional anisotropic space time fractional diffusion equation are obtained in integral form uniqueness of the solutions is studied in the framework of abstract volterra equations unimodality of propagator functions is demonstrated in the centrally symmetric case the transition from diffusion to wave propagation is discussed numerical examples are constructed

stability of reducible quadrature methods for volterra integral equations of the second kind stability analysis of reducible quadrature methods for volterra integral equations based on the test equation y t 1 int limits 0 t lambda mu t vs y s ds t geqq 0 is presented the concept of absolute stability is defined and necessary and sufficient conditions for the method to be absolutely stable for given andv are derived these conditions are illustrated for the class of methods for integral equations the main tool in our stability analysis is the theory of difference equations of poincare type

a causal and fractional all frequency wave equation for lossy media this work presents a lossy partial differential acoustic wave equation including fractional derivative terms it is derived from first principles of physics mass and momentum conservation and an equation of state given by the fractional zener stress strain constitutive relation for a derivative order in the fractional zener relation the resulting absorption k obeys frequency power laws as k 1 in a low frequency regime k 1 2 in an intermediate frequency regime and k 1 in a high frequency regime the value 1 corresponds to the case of a single relaxation process the wave equation is causal for all frequencies in addition the sound speed does not diverge as the frequency approaches infinity this is an improvement over a previously published wave equation building on the fractional kelvin voigt constitutive relation

waves in fractional zener type viscoelastic media classical wave equation is generalized for the case of viscoelastic materials obeying fractional zener model instead of hooke s law cauchy problem for such an equation is studied existence and uniqueness of the fundamental solution is proven and solution is calculated

an iterative solution strategy for boundary element equations from mixed boundary value problems abstract an iterative solution strategy is presented for mixed boundary value problems from the direct boundary element method where both dirichlet and neumann constraints are prescribed on the boundary in the mixed boundary value problem constraints are imposed upon the bem system of equations hu gq to ax b the a matrix is fully populated unsymmetric and made up of columns from h and g corresponding to the neumann and dirichlet boundary conditions respectively in this work the coefficient matrix a is rearranged to place the columns from h in the left most columns of a and the columns from g in the right most columns of a the diagonal submatrix in a containing terms from g is then reduced by elimination while the diagonal submatrix containing terms from h is retained for iteration jacobi gauss seidel adn conjugate gradient normal iterative solvers are considered convergence of the proposed solution strategy is studied using four two dimensional potential and elasticity problems

modified multilag methods for volterra functional equations linear multistep methods for ordinary differential equations in conjunction with a family of computationally efficient quadrature rules are employed to define a class of so called multilag methods for the solution of volterra integral and integro differential equations in addition modified multilag methods are proposed which have the property that the stability behavior is independent of the choice of the quadrature rules high order convergence of the methods is established in particular a special class of high order convergent methods is presented for the efficient solution of first kind volterra equations numerical experiments are reported

on the stability of volterra runge kutta methods we investigate the stability of extended runge kutta methods for volterra integral equations of the second kind as test equations we choose linear convolution equations with positive definite l 1 kernels our results are strongly connected with the contractivity of runge kutta methods for nonlinear dissipative differential equations

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations