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In this work, the ψ -Caputo fractional derivative is successfully used to define a new class of nonlinear fractional integro-differential equations. The extended Chebyshev cardinal wavelets (CCWs), as a proper set of wavelet functions, are employed to establish a computational procedure for such equations. To do this, a new operational matrix for the ψ -Riemann–Liouville fractional integral of the extended CCWs is extracted with the help of the block-pulse functions. The established approach obtains the problem solution by solving an algebraic system of nonlinear equations that is created from expanding the solution by the extended CCWs (with some unknown coefficients) and utilizing the expressed fractional operational matrix. In fact, by solving this system, the unknown coefficients and, subsequently the solution of the original problem are obtained. The convergence analysis of the proposed method is investigated. The accuracy and correctness of the scheme are illustrated by solving three examples. [ABSTRACT FROM AUTHOR]

The present study focuses on designing a multi-step technique, known as the block-by-block technique, to provide the numerical solution for a category of nonlinear fractional two-dimensional Volterra integro-differential equations. The proposed technique is a block-by-block method based on Romberg’s numerical integration formula, which simultaneously obtains highly accurate solutions at certain nodes without requiring initial starting values. The convergence analysis of the established method for the aforementioned equations is investigated using Gronwall’s inequality. Several numerical tests are presented to demonstrate the accuracy, speed, and good performance of the procedure.

This study employs a kind of non-singular variable-order fractional derivative to define a variable-order fractional version of the 2D regularized long-wave equation. The Legendre cardinal wavelets as a new family of the cardinal wavelets are introduced. A hybrid method based on the Legendre cardinal wavelets and radial basis functions is established to find the solution of this problem. This approach turns solving this equation into finding the solution of a nonlinear system of algebraic equations without utilizing discretization. The accuracy of the proposed technique is checked by solving four examples. [ABSTRACT FROM AUTHOR]

In this work, the Caputo–Hadamard time fractional version of the Rosenau equation in one and two dimensions is introduced. The logarithmic Chelyshkov functions (LCFs), as a novel family of basis functions, are introduced and used together with the classical Chelyshkov polynomials (CCPs) to established a hybrid method to solve the expressed fractional equations. For this aim, the ordinary and fractional derivative matrices associated with these basis functions are determined. To establish the desired approach by considering a hybrid expansion of the fractional problem's solution using the CCPs (for the spatial variables) and LCFs (for the temporal variable), and employing the derived derivative matrices, solving the fractional problem under consideration turns into solving an algebraic system of nonlinear equations. Some numerical examples are considered to investigate the capability and accuracy of this proposed scheme. [ABSTRACT FROM AUTHOR]

In this research, the Hadamard fractional derivative is used to define the time fractional coupled nonlinear Schrödinger–Hirota equations. The logarithmic Chebyshev cardinal functions, as a new category of cardinal functions, are introduced to build a numerical method to solve this system. To do this, the Hadamard fractional differentiation matrix of these functions is obtained. In the developed method, by considering a hybrid approximation of the problem's solution using the logarithmic Chebyshev cardinal functions (for the temporal variable) and classical Chebyshev cardinal polynomials (for the spatial variable), and employing the interpolation property of these basis functions, along with the expressed derivative matrix, solving the fractional system turns into obtaining the solution of an algebraic system of equations. Two numerical examples are investigated to acknowledge the high accuracy of the introduced procedure. • A new family of basis functions called logarithmic Chebyshev cardinal functions (LCCFs) is introduced. • Hadamard fractional differentiation matrix of these functions is obtained. • The Hadamard time fractional coupled nonlinear Schrödinger–Hirota equations are introduced. • A hybrid method based on LCCFs and classical Chebyshev cardinal polynomials is developed for this system. • The accuracy of the developed method is checked by solving some examples. [ABSTRACT FROM AUTHOR]

This paper introduces a fractal-fractional version of the nonlinear 2D advection-diffusion equation and proposes a meshless method based on the moving least squares shape functions for its numerical solution. The fractal-fractional derivative in the Atangana-Riemann-Liouville is considered to define this equation. The proposed method includes the following steps: We first approximate the fractal-fractional derivative using the finite differences method and derive a recursive algorithm by applying the θ -weighted method. Next, using the moving least squares shape functions, we expand the solution of the problem and its corresponding partial derivatives and substitute them into the recurrence formula. Finally, in accordance with the previous step, we obtain a linear system of algebraic equations which must be solved at each time step. The validity and accuracy of the method are investigated by solving some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the variable order fractional derivative in the Heydari-Hosseininia sense is employed to define a new fractional version of 2D reaction-advection-diffusion equation. An efficient and accurate hybrid method based on the shifted orthogonal Bernoulli polynomials and radial basis functions is developed for solving this equation. The presented method converts solving the problem under consideration into solving a system of algebraic equations. The applicability and accuracy of the proposed method are investigated by solving some numerical examples. [ABSTRACT FROM AUTHOR]

This study introduces a hybrid procedure based on a block-by-block scheme (created by the Gauss–Lobatto integration formula) and a set of the hybrid functions (defined by the Legendre polynomials and block-pulse functions) to solve a class of systems of mixed Volterra–Fredholm integral equations. More precisely, the proposed scheme combines the Gauss–Lobatto quadrature rule for the temporal variable and the hybrid functions for the spacial direction. In the established procedure, several values of the problem solution are elicited simultaneously, without employing any starting value for beginning. The convergence, along with the analysis of error for the method are proved. Some numerical examples are solved to show the efficiency and accuracy of the proposed strategy.

In this work, the stochastic fractional two-dimensional Sobolev equation is introduced and a collocation method is proposed to solve it. The discrete Chebyshev polynomials are used as a proper family of basis functions to establish this collocation method. Some operational matrices related to conventional and stochastic integrals, as well as fractional and ordinary derivatives of these polynomials, are extracted and successfully used in making the expressed method. More precisely, by approximating the problem solution via a finite expansion of the expressed polynomials (where the expansion coefficients are unknown) and substituting it into the stochastic fractional problem, as well as by employing the above obtained matrices, a system of linear algebraic equations is obtained. Finally, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are found by solving this system. The correctness of the established method is examined by solving some numerical examples. • The stochastic fractional two-dimensional Sobolev equation is introduced. • The orthonormal discrete Chebyshev polynomials (DCPs) are used to solve this equation. • An operational matrix of stochastic integral for the orthonormal DCPs is obtained. • Some operational matrices for derivatives of the orthonormal DCPs are extracted. • The accuracy of the proposed method is shown in some numerical examples. [ABSTRACT FROM AUTHOR]

This paper introduces the nonlinear variable-order (VO) time fractional 2D Ginzburg-Landau equation by replacing the conventional derivative with the Atangana–Baleanu–Caputo fractional derivative. An efficient moving least squares (MLS) meshfree approximation method is considered to devise an algorithm for solving this enhanced equation. To be precise, first, the finite difference scheme is used to evaluate the fractional differentiation. Then, a recurrence formula is derived by applying the θ -weighted technique. Next, the real and imaginary components of the solution are expanded in terms of meshless functions. Last, these expansions which include unknown coefficients are input in the original equation. Therefore, the equation is converted into a system of linear algebraic equations which is uncomplicated for solving by mathematical software. To verify the validity of the devised method and demonstrate its precision, several problems are put to the test. [ABSTRACT FROM AUTHOR]

In this study, the ψ-Caputo fractional derivative (as a generalization of the classical Caputo derivative where the fractional derivative is defined with respect to the function ψ) is considered to introduce a class of multi-term time fractional 2D diffusion equations. A numerical method based on the Chebyshev cardinal polynomials (CCPs) is proposed to solve this problem. In this way, a new operational matrix for the ψ-Caputo fractional derivative of the CCPs is provided. By approximating the solution of the problem by a finite series of the CCPs (with some unknown coefficients) and employing the derived fractional matrix, an algebraic system of equations is generated, which by solving it the expressed coefficients, and consequently, the problem’s solution are identified. The validity of the established method is investigated by solving some numerical examples.

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

This paper is concerned with an operational matrix method based on the shifted Legendre cardinal functions for solving the nonlinear variable-order time fractional Schrödinger equation. The variable-order fractional derivative operator is defined in the Atangana–Baleanu–Caputo sense. Through the way, a new operational matrix of variable-order fractional derivative is derived for the shifted Legendre cardinal functions and used in the established method. More precisely, the unknown solution is separated into the real and imaginary parts, and then these parts are expanded in terms of the shifted Legendre cardinal functions with undetermined coefficients. These expansions are substituted into the main equation and the generated operational matrix is utilized to extract a system of nonlinear algebraic equations. Thereafter, the yielded system is solved to obtain an approximate solution for the problem. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm that the suggested approach is high accurate in providing satisfactory results. [ABSTRACT FROM AUTHOR]

In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions of nonlinear variable-order (V-O) time fractional 2D reaction-diffusion equations. The V-O time fractional derivative is defined in the Caputo sense with Mittag-Leffler non-singular kernel of order α (x , t) ∈ (0 , 1) (known as the Atangana–Baleanu–Caputo fractional derivative). First, the V-O fractional derivative is approximated via the finite difference scheme and the theta-weighted method is utilized to derive a recursive algorithm. Then, the unknown solution of the intended problem is expanded via the 2D LWs. Finally, by applying the differentiation operational matrices in each time step, the solution of the problem is reduced to solution of a linear system of algebraic equations. In the proposed method, acceptable approximate solutions are achieved by employing only a few number of the basis functions. To illustrate the applicability, validity and accuracy of the presented wavelet method, some numerical test examples are solved. The achieved numerical results reveal that the established method is highly accurate in solving the introduced new V-O fractional model. [ABSTRACT FROM AUTHOR]

• A new class of the nonlinear variable-order (V-O) time fractional 2D telegraph equations is introduced. • An efficient and accurate meshfree method is proposed for the numerical solution of this class of problems. • The moving least squares shape functions are utilized in establishing this meshfree method. • The applicability and accuracy of the proposed approach are demonstrated by solving some numerical examples. • The proposed method can be extended for other types of V-O fractional applied problems. This paper investigates a novel version for the nonlinear 2D telegraph equation involving variable-order (V-O) time fractional derivatives in the Atangana–Baleanu–Caputo sense with Mittag–Leffler non-singular kernel. A meshfree method based on the moving least squares (MLS) shape functions is proposed for the numerical solution of this class of problems. More precisely, the V-O fractional derivatives in this model are approximated by the finite difference scheme at first. Then, the θ -weighted method is utilized to derive a recursive algorithm. Next, the solution of the problem is expanded in terms of the MLS shape functions with undetermined coefficients. Eventually, by substituting this expansion and its partial derivatives into the original equation, solution of the problem in each time step is reduced to the solution of a linear system of algebraic equations. Several numerical examples are investigated to show the applicability, validity and accuracy of the presented method. The achieved numerical results reveal that the established method is high accurate in solving such V-O fractional models. [ABSTRACT FROM AUTHOR]

The main goal of this study is to develop an efficient matrix approach for a new class of nonlinear 2D optimal control problems (OCPs) affected by variable-order fractional dynamical systems. The offered approach is established upon the shifted Chebyshev polynomials (SCPs) and their operational matrices. Through the way, a new operational matrix (OM) of variable-order fractional derivative is derived for the mentioned polynomials.The necessary optimality conditions are reduced to algebraic systems of equations by using the SCPs expansions of the state and control variables, and applying the method of constrained extrema. More precisely, the state and control variables are expanded in components of the SCPs with undetermined coefficients. Then these expansions are substituted in the cost functional and the 2D Gauss-Legendre quadrature rule is utilized to compute the double integral and consequently achieve a nonlinear algebraic equation.After that, the generated OM is employed to extract some algebraic equations from the approximated fractional dynamical system. Finally, the procedure of the constrained extremum is used by coupling the algebraic constraints yielded from the dynamical system and the initial and boundary conditions with the algebraic equation extracted from the cost functional by a set of unknown Lagrange multipliers. The method is established for three various types of boundary conditions.The precision of the proposed approach is examined through various types of test examples.Numerical simulations confirm the suggested approach is very accurate to provide satisfactory results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a computational approach based on the Chebyshev cardinal wavelets for a novel class of nonlinear stochastic differential equations characterized by the presence of variable-order fractional Brownian motion. More precisely, in the proposed approach, the solution of a nonlinear stochastic differential equation is approximated by the Chebyshev cardinal wavelets and subsequently the intended problem is transformed to a system of nonlinear algebraic equations. In this way, the nonlinear terms are significantly reduced, due to the cardinal property of the basis functions used. The convergence analysis of the expressed method is theoretically investigated. Moreover, the reliability and applicability of the approach are experimentally examined through the numerical examples. In addition, the presented method is implemented for some famous stochastic models, such as stochastic logistic problem, stochastic population growth model, stochastic Lotka–Volterra problem, stochastic Brusselator problem, stochastic Duffing-Van der Pol oscillator problem and stochastic pendulum model. As another new finding, a procedure is established for constructing the variable-order fractional Brownian motion. Indeed, the standard Brownian motion together with the block pulse functions and the hat functions are utilized for generating the variable-order fractional Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, a new computational method is proposed to solve a class of nonlinear stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm). The method is based on a new class of orthogonal wavelets, namely the Chebyshev cardinal wavelets. These new basis functions have many useful properties such as orthogonality, spectral accuracy, and cardinality. The operational matrices of the derivative and the integration of these new wavelets are derived and used in the implementation of the proposed method. Based on the interpolation property of these new wavelets, a new algorithm is presented for computing nonlinear terms in such problems. The main advantage of the proposed method is the reduction of the problem to a simpler one, which consists of solving a system of nonlinear algebraic equations. The convergence and error analysis of the established method are investigated in the Sobolev space. Moreover, the reliability and applicability of the proposed method are demonstrated by solving several numerical examples. Finally, the application of the proposed method is illustrated by solving two well-known fractional stochastic models in the mathematical finance and the mathematical ecology. [ABSTRACT FROM AUTHOR]

In this paper, two important subjects are investigated. First, a fruitful family of functions with the interpolation property called the orthonormal piecewise fractional Chebyshev cardinal functions are generated, and a formulation for their fractional derivative matrix is provided. Second, these functions together with their expressed matrix are used to construct a numerical method for fractional Ginzburg–Landau equation with a non-smooth solution. In fact, the proposed method approximates the problem solution by the expressed fractional functions (in the temporal direction) and classical Chebyshev cardinal polynomials (in the space direction). This method is able to get highly accurate solutions for the problem under investigation if its solution is in the piecewise form or includes terms with fractional powers, or includes both of them. The ability and validity of the method are examined by solving some numerical examples. • Fractional piecewise Chebyshev cardinal functions are introduced. • A formulation for computing fractional derivative of these functions is derived. • Non-smooth solution for fractional Ginzburg-Landau equation is obtained. • The accuracy of the proposed method is shown in some numerical examples. [ABSTRACT FROM AUTHOR]

In this work, the distributed-order time fractional version of the Schrödinger problem is defined by replacing the first order derivative in the classical problem with this kind of fractional derivative. The Caputo fractional derivative is employed in defining the used distributed fractional derivative. The orthonormal piecewise Jacobi functions as a novel family of basis functions are defined. A new formulation for the Caputo fractional derivative of these functions is derived. A numerical method based upon these piecewise functions together with the classical Jacobi polynomials and the Gauss–Legendre quadrature rule is constructed to solve the introduced problem. This method converts the mentioned problem into an algebraic problem that can easily be solved. The accuracy of the method is examined numerically by solving some examples. • The distributed-order time fractional version of the Schrödinger equation is defined. • The orthonormal piecewise Jacobi functions as a new class of basis functions are defined. • A new formulation for the Caputo fractional derivative of these piecewise functions is derived. • The Jacobi polynomials and the piecewise Jacobi functions are employed mutually to solve this problem. • The accuracy of the proposed method is investigated in some numerical examples. [ABSTRACT FROM AUTHOR]

In this work, a category of delay distributed-order time fractional fourth-order sub-diffusion equations is investigated. The Chebyshev cardinal polynomials (as a proper class of basis functions) are employed to make an appropriate methodology for these problems. To this end, some matrix relationships regarding the distributed-order fractional differentiation (in the Caputo kind) of these polynomials are extracted and applied in generating the desired approach. The provided method converts solving these problems into obtaining the solution of systems of algebraic equations. The reliability of the technique is evaluated by solving three examples. • The distributed-order time fractional fourth-order sub-diffusion equations is defined. • The Chebyshev cardinal polynomials are employed to solve these problems. • A distributed-order fractional derivative matrix is extracted. • The accuracy of the proposed method is investigated in some numerical examples. [ABSTRACT FROM AUTHOR]

Highlights: [•] Singular fractional Volterra integro-differential equations. [•] In this paper it is given a method for solving singular fractional volterra integro-differential equations. [•] This method is focussed on a special family of wavelets (Chebyshev). [•] Chebyshev wavelets are some special wavelets having both the many advantages of wavelets and the Chebyshev polynomials. [•] Operational matrix: the fractional derivatives when applied to some polynomials enable to define some operational matrices. [•] These matrices will be explicitly computed. [•] This method is applied to some examples and the error is explicitly computed. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the two-dimensional Legendre wavelets are applied for numerical solution of the fractional Poisson equation with Dirichlet boundary conditions. In this way, a new operational matrix of fractional derivative for the Legendre wavelets is derived and then this operational matrix has been employed to obtain the numerical solution of the above-mentioned problem. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence of the two-dimensional Legendre wavelets expansion is investigated. Also the power of this manageable method is illustrated. [Copyright &y& Elsevier]

This paper presents a numerical technique for solving the variable-order fractional extended Fisher–Kolmogorov equation. The method suggested to solve this problem is based on the orthonormal shifted discrete Legendre polynomials and the collocation method. First, we expand the unknown solution of the problem using the these polynomialss. Also, we approximate the second- and fourth-order classical derivatives, as well as the variable-order fractional derivatives by these basis functions. Then, we substitute these approximations in the equation. Next, we utilize the classical and fractional derivative matrices together with the collocation method to convert the main equation into a system containing nonlinear algebraic equations. We show the correctness of the proposed scheme by providing several numerical examples. [ABSTRACT FROM AUTHOR]

In this study, a kind of piecewise fractional derivatives based on the Caputo fractional derivative is used to define a novel category of fractional optimal control problems. The piecewise Chebyshev cardinal functions as an appropriate family of basis functions are considered to construct a numerical method for solving such problems. The classical and piecewise fractional derivative matrices of these basis functions are derived and used in constructing the proposed technique. The established scheme transforms obtaining the solution of such problems into finding the solution of algebraic systems of equations by approximating the state and control variables using the mentioned basis functions. The accuracy of the expressed approach is investigated by solving some examples. [ABSTRACT FROM AUTHOR]

• A new set of basis functions called the piecewise Chebyshev cardinal functions is introduced. • These basis functions possess many useful properties, such as orthogonality, cardinality and spectral accuracy. • The operational matrix of fractional integration is obtained for these functions. • A direct matrix method is developed to investigate a class of constrained fractional optimal control problems. In this paper, a new set of basis functions called the piecewise Chebyshev cardinal functions is generated to investigate a class of constrained fractional optimal control problems. These basis functions possess many useful properties, such as orthogonality, cardinality and spectral accuracy. The fractional integral matrix of these functions is obtained. A direct scheme based on the these basis functions together with their fractional integral matrix is developed for solving the problem under consideration. The established method transforms solving the original problem into solving a constrained minimization problem by approximating the state and control variables in terms of the piecewise Chebyshev cardinal functions. Some numerical examples are given to show the efficiency of the proposed technique. [ABSTRACT FROM AUTHOR]

• A new fractal-fractional (FF) derivative operator with Mittag-Leffler function as its kernel is introduced. • The orthonormal shifted discrete Chebyshev polynomials are generated. • Some new operational matrices are derived for the mentioned basis polynomials. • The FF model of the coupled nonlinear Schrodinger-Boussinesq equations is defined. • A computational matrix method is developed to solve the defined system. In this paper, a novel fractal-fractional derivative operator with Mittag-Leffler function as its kernel is introduced. Using this differentiation, the fractal-fractional model of the coupled nonlinear Schrödinger-Boussinesq system is defined. The orthonormal shifted discrete Chebyshev polynomials are generated and used for constructing a computational matrix method to solve the defined system. In the established method, using the matrices of the ordinary and fractal-fractional differentiations of these polynomials, the fractal-fractional system transformed into a system of algebraic equations, which is solved readily. Practicability and precision of the method are examined by solving two numerical examples. [ABSTRACT FROM AUTHOR]

This paper introduces a novel class of nonlinear optimal control problems generated by dynamical systems involved with variable-order fractional derivatives in the Atangana–Baleanu–Caputo sense. A computational method based on the Chebyshev cardinal functions and their operational matrix of variable-order fractional derivative (which is generated for the first time in the present study) is proposed for the numerical solution of this class of problems. The presented method is based on transformation of the main problem to solving system of nonlinear algebraic equations. To do this, the state and control variables are expanded in terms of the Chebyshev cardinal functions with unknown coefficients, then the cardinal property of these basis functions together with their operational matrix are employed to generate a constrained extremum problem, which is solved by the Lagrange multipliers method. The applicability and accuracy of the established method are investigated through some numerical examples. The reported results confirm that the established scheme is highly accurate in providing acceptable results. [ABSTRACT FROM AUTHOR]

Glioblastoma is the most common and deadly primary brain tumor in adults. To optimize the treatment strategies, it is essential to understand the tumor growth dynamics in different periods. In this study, we use image processing techniques to combine the available early-stage imaging data and applied a fractional reaction–diffusion equation to predict the human glioblastoma multiforme tumor growth. We consider the heterogeneity of the brain tissue by assigning different diffusion coefficients for the three regions of the human brain. A meshfree method based on the thin plate spline radial basis function is used for the numerical solution of nonlinear time fractional Proliferation-Invasion equation. The results show that the proposed model has a better fit with the experimental data. The prediction of tumor growth at any desired time with no need to repeated imaging is another advantages of the model which could reduce the side effects and cost of diagnostic and therapeutic methods. The model can also incorporate the effects of various treatment modalities such as hyperthermia, radiation, and surgery on tumor growth. Furthermore, it can enable the use of patient-specific characteristics in diagnosis and treatment and facilitate the development of personalized medicine approaches. [ABSTRACT FROM AUTHOR]

The epidemiological aspects of the viral dynamic of the SARS-CoV-2 have become increasingly crucial due to major questions and uncertainties around the unaddressed issues of how corpse burial or the disposal of contaminated waste impacts nearby soil and groundwater. Here, a theoretical framework base on a meshless algorithm using the moving least squares (MLS) shape functions is adopted for solving the time-fractional model of the viral diffusion in and across three different environments including water, tissue, and soil. Our computations predict that by considering the α (order of fractional derivative) best fit to experimental data, the virus has a traveling distance of 1 m m in water after 22, regardless of the source of contamination (e.g., from tissue or soil). The outcomes and extrapolations of our study are fundamental for providing valuable benchmarks for future experimentation on this topic and ultimately for the accurate description of viral spread across different environments. In addition to COVID-19 relief efforts, our methodology can be adapted for a wide range of applications such as studying virus ecology and genomic reservoirs in freshwater and marine environments. [ABSTRACT FROM AUTHOR]

Fractional differential equations (FDEs) offer numerous capabilities for modeling unusual phenomena. So, the study of these models is essential. This paper proposes an efficient meshless technique for obtaining the numerical solution of a space fractional diffusion model with Caputo derivative type. Typically, in a meshless processes based on moving Kriging (MK) interpolation, the MK technique is used to calculate the shape functions and their derivatives with positive integer order against space variables to discretize the governing equation in space variables. However, in this study, we employ the Taylor series of MK interpolation's correlation function to enhance the shape function derivatives for arbitrary order 0. 5 < α < 2. Moreover, the finite difference technique is employed to discretize the model in the time dimension and its characterized by unconditional stability and a rate of convergence of O (τ). This approach converts the primary problem into a system of linear algebraic equations. Finally, we present several examples in one and two dimensions and compare the results with other well-known schemes to show the capability and accuracy of this approach. [ABSTRACT FROM AUTHOR]

The glioblastoma multiforme (GBM) is the fast-growing and aggressive type of tumor of the brain or spinal cord that is associated with high morbidity and mortality. Therefore, accurate knowledge of the tumor growth process in different time intervals seems necessary to adopt optimal treatment methods. In this study, with a combination of available early stages imaging data and using image processing methods, a time fractional reaction–diffusion equation based on the moving least squares method was implemented for predicting the growth of the GBM tumor model implemented in the mouse brain. The obtained results demonstrate that the Proliferation-Invasion model implemented by a time fractional form shows more agreement with the experimental results. As a result, using a fractional derivative of order α = 0. 9 results in the lowest prediction error of the tumor surface (less than 1 %). In addition, to reducing the cost and side effects of theranostic methods, the presented model will have the possibility to consider the effects of therapeutic factors, such as hyperthermia, radiography, and surgery, on tumor growth. The ability to use the patient's characteristics in diagnostic and treatment considerations and the development of personalized medicine can also be considered one of the capabilities of the presented model. • The tumor growth is predicted via the MR images processing and mathematical model. • It shown that fractional PI model is more agreement with the experiment results. • Fractional model makes the tumor growth predictable at any time without any reimaging. • The patient's characteristics in theranostics is one of the capabilities of the model. • A meshless method is proposed for nonlinear fractional reaction-diffusion equation. [ABSTRACT FROM AUTHOR]