Muntz Polynomials Research Articles

Approximate Solution for a Fuzzy Fractional Multi-term Differential Equation by Müntz Polynomials

In this study, we present a new approach for the numerical solution of a fuzzy fractional multi term differential equation under the Caputo fractional derivative which orders in [0, 1]. To do this, collocation method based on Muntz polynomials are used. As in the interval [0, 1] the introduced Muntz polynomials and their fractional derivatives of order $$0 \le \alpha \le 1$$ are not negative; therefore, they are suitable bases for fuzzy approximate. Finally, some examples were included to confirm the applicability, accuracy, and efficiency of the proposed method.

Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling

B-splines $$B_{q}$$ of quaternionic order q, for short quaternionic B-splines, are quaternion-valued piecewise Muntz polynomials whose scalar parts interpolate the classical Schoenberg splines $$B_{n}$$ , $$n\in \mathbb {N}$$ , with respect to degree and smoothness. They in general do not satisfy the interpolation property $$B_{q}(n) = \delta _{n,0}$$ , $$n\in \mathbb {Z}$$ . However, the application of the interpolation filter $$(\sum \nolimits _{k\in \mathbb {Z}} \widehat{B_q}(\xi +2 \pi k))^{-1}$$ —if well-defined—in the frequency domain yields a cardinal fundamental spline of quaternionic order that satisfies the interpolation property. We handle the ambiguity of the quaternion-valued exponential function appearing in the denominator of the interpolation filter and relate the filter to interesting properties of a quaternionic Hurwitz zeta function and the existence of complex quaternionic inverses. Finally, we show that the cardinal fundamental splines of quaternionic order fit into the setting of Kramer’s Lemma and allow for a family of sampling, respectively, interpolation series.

A robust kernel-based solver for variable-order time fractional PDEs under 2D/3D irregular domains

This study presents a robust kernel-based collocation method (KBCM) for solving multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method, Radial basis functions (RBFs) and Muntz polynomials basis (MPB) are implemented to discretize the spatial and temporal derivative terms in the VOTFPDEs, respectively. Due to the properties of the RBFs, the spatial discretization in the proposed method is mathematically simple and truly meshless, which avoids troublesome mesh generation for high-dimensional problems involving irregular geometries. Due to the properties of the MPB, only few temporal discretization is required to achieve the satisfactory accuracy. Numerical efficiency of the proposed method is investigated under several typical examples.

A collocation method for solving the fractional calculus of variation problems

In this paper we use a family of Muntz polynomials and a computational technique based on the collocation method to solve the calculus variation problem. This approach is utilizedto reduce the solution of linear and nonlinear fractional order dierential equations to the solution of a system of algebraic equations. Thus we can obtain a good approximation evenby using a smaller of collocation points.

A fractional spectral method with applications to some singular problems

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Muntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x 1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.

A New Extended Padé Approximation and Its Application

We extend ordinary Pad&#xe9; approximation, which is based on a set of standard polynomials as <svg style="vertical-align:-2.30685pt;width:85.125px;" id="M1" height="16.487499" version="1.1" viewBox="0 0 85.125 16.487499" width="85.125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,13.55)"><path id="x7B" d="M293 -169q-145 0 -145 128q0 35 4 106t3 104q0 17 -2 28t-10 25.5t-28 23t-51 10.5v29q31 2 51 10.5t28 23t10 25.5t2 28q0 31 -4 101t-3 104q0 132 145 132v-28q-51 -7 -67.5 -30t-16.5 -63q0 -32 6.5 -99.5t6.5 -100.5q0 -97 -83 -115v-4q83 -20 83 -117&#xA;q0 -34 -6.5 -99.5t-6.5 -96.5q0 -41 16.5 -65.5t67.5 -31.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,5.961,13.55)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,14.12,13.55)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,20.818,13.55)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g><g transform="matrix(.017,-0,0,-.017,30.32,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.035,13.55)"><path id="x2026" d="M160 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM485 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16zM806 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5&#xA;t16 -40.5q0 -26 -15.5 -42t-41.5 -16z" /></g><g transform="matrix(.017,-0,0,-.017,56.345,13.55)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,63.06,13.55)"><use xlink:href="#x1D465"/></g> <g transform="matrix(.012,-0,0,-.012,72.575,5.388)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g> <g transform="matrix(.017,-0,0,-.017,79.175,13.55)"><path id="x7D" d="M283 255q-31 -2 -51 -10.5t-28 -23.5t-10 -27t-2 -30q0 -28 4 -97.5t4 -107.5q0 -128 -146 -128v28q50 7 67.5 31t17.5 63q0 33 -7 100t-7 98q0 96 84 117v4q-84 20 -84 115q0 34 7 102.5t7 98.5q0 40 -17.5 63t-67.5 30v28q28 0 50 -4t45.5 -16.5t37 -40t13.5 -68.5&#xA;q0 -37 -4 -106.5t-4 -98.5q0 -18 2 -30t10 -27t28 -23.5t51 -10.5v-29z" /></g> </svg>, to a new extended Pad&#xe9; approximation (M&#xfc;ntz Pad&#xe9; approximation), based on the general basic function set <svg style="vertical-align:-2.30685pt;width:128px;" id="M2" height="19.625" version="1.1" viewBox="0 0 128 19.625" width="128" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,16.7)"><use xlink:href="#x7B"/></g><g transform="matrix(.017,-0,0,-.017,5.961,16.7)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,14.12,16.7)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,20.818,16.7)"><use xlink:href="#x1D465"/></g> <g transform="matrix(.012,-0,0,-.012,30.338,8.537)"><path id="x1D706" d="M529 97q-70 -109 -136 -109q-41 0 -56 94q-23 144 -37 284q-38 -88 -99 -202.5t-93 -156.5q-26 -8 -76 -19l-9 21q71 78 145.5 193t124.5 232q-5 84 -15 128q-12 55 -29.5 75.5t-42.5 20.5q-21 0 -45 -13l-8 24q16 17 46 30t55 13q43 0 70 -46.5t40 -169.5&#xA;q27 -249 51 -392q7 -46 23 -46q24 0 70 60z" /></g> <g transform="matrix(.017,-0,0,-.017,37.525,16.7)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,44.223,16.7)"><use xlink:href="#x1D465"/></g> <g transform="matrix(.012,-0,0,-.012,53.738,8.537)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g><g transform="matrix(.012,-0,0,-.012,59.449,8.537)"><use xlink:href="#x1D706"/></g> <g transform="matrix(.017,-0,0,-.017,66.638,16.7)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,73.335,16.7)"><use xlink:href="#x2026"/></g><g transform="matrix(.017,-0,0,-.017,92.663,16.7)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,99.36,16.7)"><use xlink:href="#x1D465"/></g> <g transform="matrix(.012,-0,0,-.012,108.875,8.537)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.012,-0,0,-.012,114.848,8.537)"><use xlink:href="#x1D706"/></g> <g transform="matrix(.017,-0,0,-.017,122.038,16.7)"><use xlink:href="#x7D"/></g> </svg>&#x2009;&#x2009;<svg style="vertical-align:-2.3205pt;width:76.4375px;" id="M3" height="15.1125" version="1.1" viewBox="0 0 76.4375 15.1125" width="76.4375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,12.162)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g><g transform="matrix(.017,-0,0,-.017,18.812,12.162)"><path id="x3C" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" /></g><g transform="matrix(.017,-0,0,-.017,33.516,12.162)"><use xlink:href="#x1D706"/></g><g transform="matrix(.017,-0,0,-.017,47.625,12.162)"><path id="x2264" d="M531 71l-474 214v50l474 215v-56l-416 -184l416 -183v-56zM531 -40h-474v50h474v-50z" /></g><g transform="matrix(.017,-0,0,-.017,62.329,12.162)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,70.488,12.162)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> (the particular case of M&#xfc;ntz polynomials) using general Taylor series (based on fractional calculus) with error convergency. The importance of this extension is that the ordinary Pad&#xe9; approximation is a particular case of our extended Pad&#xe9; approximation. Also the parameterization (<svg style="vertical-align:-0.1638pt;width:9.5124998px;" id="M4" height="12.4375" version="1.1" viewBox="0 0 9.5124998 12.4375" width="9.5124998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D706"/></g> </svg> is the corresponding parameter) of new extended Pad&#xe9; approximation is an important subject which, obtaining the optimal value of this parameter, can be a good question for a new research.

Bilinear optimality constraints for the cone of positive polynomials

For a proper cone $${{\mathcal K}\subset\mathbb{R}^n}$$ and its dual cone $${{\mathcal K}^*}$$ the complementary slackness condition $${\langle{\rm {\bf x}},{\rm {\bf s}}\rangle=0}$$ defines an n-dimensional manifold $${C({\mathcal K})}$$ in the space $${{\mathbb R}^{2n}}$$. When $${{\mathcal K}}$$ is a symmetric cone, points in $${C({\mathcal K})}$$ must satisfy at least n linearly independent bilinear identities. This fact proves to be useful when optimizing over such cones, therefore it is natural to look for similar bilinear relations for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for points in $${C({\mathcal K})}$$. We examine several well-known cones, in particular the cone of positive polynomials $${{\mathcal P}_{2n+1}}$$ and its dual, and show that there are exactly four linearly independent bilinear identities which hold for all $${({\rm {\bf x}},{\rm {\bf s}})\in C({\mathcal P}_{2n+1})}$$, regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential polynomials. We prove similar results for Muntz polynomials.

Fundamental Sets of Fractal Functions

Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Muntz polynomials that successfully generalize classical Muntz polynomials. The parameters of the fractal Muntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Muntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains.

Gaussian-type Quadrature Rules for Müntz Systems

A method for constructing the generalized Gaussian quadrature rules for Muntz polynomials on $(0,1)$ is given. Such quadratures possess several properties of the classical Gaussian formulae (for polynomial systems), such as positivity of the weights, rapid convergence, etc. They can be applied to the wide class of functions, including smooth functions, as well as functions with end-point singularities, such as those in boundary-contact value problems, integral equations with singular kernels, complex analysis, potential theory, etc. The constructive method is based on an application of orthogonal Muntz polynomials introduced by Badalyan [{\it Akad. Nauk Armyan. SSR. Izv. Fiz.-Mat. Estest. Tehn. Nauk}, 8 (1955), pp. 1--28; 9 (1956), pp. 3--22 (Russian; Armenian summary)] and studied intensively by Borwein, Erdelyi, and Zhang [{\it Trans. Amer. Math. Soc}., 342 (1994), pp. 523--542], as well as by Milovanovic on a numerical procedure for evaluation of such polynomials with high precision [{\it Muntz orthogonal polynomials and their numerical evaluation}, in Applications and Computation of Orthogonal Polynomials, Internat. Ser. Numer. Math. 131, W. Gautschi, G. H. Golub, and G. Opfer, eds., Birkhauser, Basel, 1999, pp. 179--194]. The method is numerically stable. Some numerical examples are included.

The full Markov-Newman inequality for Müntz polynomials on positive intervals

For a function f defined on an interval [a, b] let ∥f∥ [a,b] := sup{|f(x)|: x ∈ [a,b]}. The principal result of this paper is the following Markov-type inequality for Muntz polynomials. Theorem. Let n > 1 be an integer. Let λ 0 , λ 1 ,...,λ n be n + 1 distinct real numbers. Let 0 < a < b. Then formula math. formula math. where the supremum is taken for all Q ∈ span{x λ 0,x λ 1,.., x λ n} (the span is the linear span over R).

Piecewise Monotone Interpolation and Approximation with Muntz Polynomials

The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of “Muntz polynomials” on a real interval is demonstrated: (i) approximation of a continuous function by a “copositive” Muntz polynomial; (ii) approximation of a continuous function by a “comonotone” Muntz polynomial; (iii) approximation of a continuous function with a monotone

Piecewise monotone interpolation and approximation with Muntz polynomials

The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of “Muntz polynomials” on a real interval is demonstrated: (i) approximation of a continuous function by a “copositive” Muntz polynomial; (ii) approximation of a continuous function by a “comonotone” Muntz polynomial; (iii) approximation of a continuous function with a monotone kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.