In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legen...In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices.Further,operational matrices are constructed using variable order differentiation and integration.We are finding the operationalmatrices of variable order differentiation and integration by omitting the discretization of data.With the help of aforesaid matrices,considered FDEs are converted to algebraic equations of Sylvester type.Finally,the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions.Some examples are given to check the proposed method’s accuracy and graphical representations.Exact and numerical solutions are also compared in the paper for some examples.The efficiency of the method can be enhanced further by increasing the scale level.展开更多
This paper proposes a new set of 3D rotation scaling and translation invariants of 3D radially shifted Legendre moments. We aim to develop two kinds of transformed shifted Legendre moments： a 3D substituted radial sh...This paper proposes a new set of 3D rotation scaling and translation invariants of 3D radially shifted Legendre moments. We aim to develop two kinds of transformed shifted Legendre moments： a 3D substituted radial shifted Legendre moments （3DSRSLMs） and a 3D weighted radial one （3DWRSLMs）. Both are centered on two types of polynomials. In the first case, a new 3D ra- dial complex moment is proposed. In the second case, new 3D substituted/weighted radial shifted Legendremoments （3DSRSLMs/3DWRSLMs） are introduced using a spherical representation of volumetric image. 3D invariants as derived from the sug- gested 3D radial shifted Legendre moments will appear in the third case. To confirm the proposed approach, we have resolved three is- sues. To confirm the proposed approach, we have resolved three issues： rotation, scaling and translation invariants. The result of experi- ments shows that the 3DSRSLMs and 3DWRSLMs have done better than the 3D radial complex moments with and without noise. Sim- ultaneously, the reconstruction converges rapidly to the original image using 3D radial 3DSRSLMs and 3DWRSLMs, and the test of 3D images are clearly recognized from a set of images that are available in Princeton shape benchmark （PSB） database for 3D image.展开更多
基金Supporting Project No.(PNURSP2022R 14),Princess Nourah bint A bdurahman University,Riyadh,Saudi Arabia.
文摘In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices.Further,operational matrices are constructed using variable order differentiation and integration.We are finding the operationalmatrices of variable order differentiation and integration by omitting the discretization of data.With the help of aforesaid matrices,considered FDEs are converted to algebraic equations of Sylvester type.Finally,the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions.Some examples are given to check the proposed method’s accuracy and graphical representations.Exact and numerical solutions are also compared in the paper for some examples.The efficiency of the method can be enhanced further by increasing the scale level.
文摘This paper proposes a new set of 3D rotation scaling and translation invariants of 3D radially shifted Legendre moments. We aim to develop two kinds of transformed shifted Legendre moments： a 3D substituted radial shifted Legendre moments （3DSRSLMs） and a 3D weighted radial one （3DWRSLMs）. Both are centered on two types of polynomials. In the first case, a new 3D ra- dial complex moment is proposed. In the second case, new 3D substituted/weighted radial shifted Legendremoments （3DSRSLMs/3DWRSLMs） are introduced using a spherical representation of volumetric image. 3D invariants as derived from the sug- gested 3D radial shifted Legendre moments will appear in the third case. To confirm the proposed approach, we have resolved three is- sues. To confirm the proposed approach, we have resolved three issues： rotation, scaling and translation invariants. The result of experi- ments shows that the 3DSRSLMs and 3DWRSLMs have done better than the 3D radial complex moments with and without noise. Sim- ultaneously, the reconstruction converges rapidly to the original image using 3D radial 3DSRSLMs and 3DWRSLMs, and the test of 3D images are clearly recognized from a set of images that are available in Princeton shape benchmark （PSB） database for 3D image.
