An existence result for a class of second order evolution equations of mixed type

We give an existence and uniqueness result for a linear abstract evolution equation of second order with some coefficient in front of the second temporal derivative which may degenerate to zero and change sign.     

On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order

In this paper, the authors investigate the growth of solutions of a class of higher order linear differential equations

f(k)+Ak−1f(k−1)+?+A₀f=0     

Solution of high‐order linear Fredholm integro‐differential equations with piecewise intervals

In this study, a practical matrix method is presented to find an approximate solution for high‐order linear Fredholm integro‐differential equations with piecewise intervals under the initial boundary conditions in terms of Taylor polynomials. The method converts the integro differential equation to a matrix equation, which corresponds to a system of linear algebraic equations. Error analysis and illustrative examples are included to demonstrate the validity and applicability of the technique. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 27: 1327–1339, 2011     

A Hermite collocation method for the approximate solutions of high‐order linear Fredholm integro‐differential equations

In this study, a Hermite matrix method is presented to solve high‐order linear Fredholm integro‐differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011     

A difference method for solving parabolic equations of order 2n

In this paper, a finite difference method is used to approximate for the solution of the parabolic partial differential equation of order 2n and error of the method is determined. The resulting system is solved by efficient implicit iterations. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method.     

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.     

Distributed order calculus and equations of ultraslow diffusion

We consider equations of the form

     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

Multiresolution expansion, approximation order and quasiasymptotic behavior of tempered distributions

Multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces Sr(R), r∈N₀. For a function h, which is smooth enough and of appropriate decay, it is shown that the derivatives of its projections to the corresponding spaces Vj, j∈Z, in a regular multiresolution analysis of L²(R), denoted by hj, multiplied by a polynomial weight converge in sup norm, i.e., hj→h in Sr(R) as j→∞. Analogous result for tempered distributions is obtained by duality arguments. The analysis of the approximation order of the projection operator within the framework of the theory of shift-invariant spaces gives a further refinement of the results. The order of approximation is measured with respect to the corresponding space of test functions. As an application, we give Abelian and Tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity.     

A new operational vector approach for time-fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time-fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block-pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.     

Second order linear difference equations

We provide the explicit solution of a general second order linear difference equation via the computation of its associated Green function. This Green function is completely characterized and we obtain a closed expression for it using functions of two–variables, that we have called Chebyshev functions due to its intimate relation with the usual one–variable Chebyshev polynomials. In fact, we show that Chebyshev functions become Chebyshev polynomials if constant coefficients are considered.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Laguerre matrix method with the residual error estimation for solutions of a class of delay differential equations

In this study, a practical matrix method based on Laguerre polynomials is presented to solve the higher‐order linear delay differential equations with constant coefficients and functional delays under the mixed conditions. Also, an error analysis technique based on residual function is developed and applied to some problems to demonstrate the validity and applicability of the method. In addition, an algorithm written in Matlab is given for the method. Copyright © 2013 John Wiley & Sons, Ltd.     

A new collocation method for solution of mixed linear integro-differential-difference equations

Numerical solution of mixed linear integro-differential-difference equation is presented using Chebyshev collocation method. The aim of this article is to present an efficient numerical procedure for solving mixed linear integro-differential-difference equations. Our method depends mainly on a Chebyshev expansion approach. This method transforms mixed linear integro-differential-difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple10.     

The growth order of solutions of systems of complex difference equations

Using Nevanlinna theory of the value distribution of meromorphic functions and Wiman-Valiron theory of entire functions, we investigate the problem of growth order of solutions of a type of systems of difference equations, and extend some results of the growth order of solutions of systems of differential equations to systems of difference equations.     

On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p$p$ and a parameter λ>0$\lambda >0$. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂Rⁿ$B\subset {\mathbb{R}}^n$. Regularity of solutions strictly depends on the power p$p$ and the parameter λ$\lambda$. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.     

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form

W(t)=t^αe^-te^Att^Bt^B*e^A*t,