Solution of a Class of Partial Differential Equations with Initial Conditions

1974 ◽  
Vol 5 (6) ◽  
pp. 918-919
Author(s):  
L. Billard ◽  
Norman C. Severo
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
Partial Differential

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

scholarly journals Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Weishi Yin ◽  
Fei Xu ◽  
Weipeng Zhang ◽  
Yixian Gao
Keyword(s):  
Asymptotic Expansion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Power Series Method ◽  
Time Space ◽  
Residual Power Series ◽  
Asymptotic Expansion Of Solutions

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.

scholarly journals Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 113-126
Author(s):  
Xuping Zhang ◽  
Qiyu Chen ◽  
Yongxiang Li
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Linear Part ◽  
Fractional Power ◽  
Interpolation Spaces ◽  
Partial Differential ◽  
Nonlocal Initial Conditions ◽  
Spatial Derivatives

AbstractThis paper is devoted to study the existence and regularity of mild solutions in some interpolation spaces for a class of functional partial differential equations with nonlocal initial conditions. The linear part is assumed to be a sectorial operator in Banach space X. The fractional power theory and α-norm are used to discuss the problem so that the obtained results can be applied to equations with terms involving spatial derivatives. Moreover, we present an example to illustrate the application of main results.

scholarly journals The method of upper, lower solutions and monotone iterative scheme for higher order hyperbolic partial differential equations

1989 ◽  
Vol 47 (1) ◽  
pp. 153-170 ◽  
Author(s):  
Ravi P. Agarwal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Scheme ◽  
Initial Conditions ◽  
Upper And Lower Solutions ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Monotone Iterative ◽  
Variation Of Parameters ◽  
Variation Of Parameters Formula

AbstractUniformly monotone convergent iterative methods for obtaining multiple solutions of (n + m)th order hyperbolic partial differential equations together with initial conditions are discussed. Appropriate partial differential inequalities which connect upper and lower solutions, and variation of parameters formula is developed.

scholarly journals On the influence of the initial data in a combustion problem

1980 ◽  
Vol 22 (2) ◽  
pp. 193-209 ◽  
Author(s):  
K. K. Tam
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Coupled System ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Approximate Expressions ◽  
Combustion Problem

AbstractThe combustion of a material can be modelled by two coupled parabolic partial differential equations for the temperature and concentration of the material. This paper deals with properties of the solution of these equations inside a cylinder or a sphere and under given initial conditions. Bounds for the variation of the temperature with the initial conditions are first established by considering a decoupled form of the equations. Then the coupled system is used to obtain approximate expressions for the temporal evolution of temperature and concentration.

scholarly journals Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases

2021 ◽  
Vol 5 (3) ◽  
pp. 106
Author(s):  
Muhammad Bhatti ◽  
Md. Rahman ◽  
Nicholas Dimakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Matrix Equation ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Initial Guess ◽  
Partial Differential

A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product of variable coefficients ai(t) and B-polys Bi(x) has been employed. Additionally, the variable quantities in the anticipated solution are determined using the Galerkin method for minimizing errors. Before the minimization process is to take place, the NPDE is converted into an operational matrix equation which, when inverted, yields values of the undefined coefficients in the expected solution. The nonlinear terms of the NPDE are combined in the operational matrix equation using the initial guess and iterated until converged values of coefficients are obtained. A valid converged solution of NPDE is established when an appropriate degree of B-poly basis is employed, and the initial conditions are imposed on the operational matrix before the inverse is invoked. However, the accuracy of the solution depends on the number of B-polys of a certain degree expressed in multidimensional variables. Four examples of NPDE have been worked out to show the efficacy and accuracy of the two-dimensional B-poly technique. The estimated solutions of the examples are compared with the known exact solutions and an excellent agreement is found between them. In calculating the solutions of the NPD equations, the currently employed technique provides a higher-order precision compared to the finite difference method. The present technique could be readily extended to solving complex partial differential equations in multivariable problems.

scholarly journals Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

2021 ◽  
Vol 5 (4) ◽  
pp. 208
Author(s):  
Muhammad I. Bhatti ◽  
Md. Habibur Rahman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Order ◽  
Initial Conditions ◽  
Operational Matrix ◽  
Basis Sets ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
The Galerkin Method

A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields.

Does a Real Material Behave Fractionally? Applications of Fractional Differential Operators to the Damped Structure Borne Sound in Viscoelastic Solids

2003 ◽  
Vol 11 (03) ◽  
pp. 451-489 ◽  
Author(s):  
Bodo Nolte ◽  
Sigmar Kempfle ◽  
Ingo Schäfer
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Initial Conditions ◽  
Time Derivative ◽  
Initial Boundary ◽  
Memory Effects ◽  
Partial Differential ◽  
Damping Behavior ◽  
Resonance Frequencies

Time dependent analysis of the dynamic damped behavior of continua are mathematically modelled by partial differential equations. One obtains uniqueness, existence and stability (well posed problems) by the implementation of the correct initial boundary conditions. However, by taking memory effects into consideration, any change in the past of the system changes the future dynamic behavior. Classical damping descriptions fail when describing the behavior of many materials, like teflon. This is because in classical theory the operators are local ones. The implementation of fractional time derivatives into the partial differential equations is an alternative technique to overcome these problems. Thereby the time derivative operator is a global one, memory effects in structure borne sound can be calculated. In this paper the theory of fractional time derivative operators and their application in continuum mechanics is presented. The main result when using this method for damping behavior is that a global operator is needed which takes the whole history into account. We call this theory the functional calculus method instead of the well-known fractional calculus with the use of initial conditions. In order to show the efficiency of this method the calculated impulse response of a viscoelastic rod is compared with measurements. It is shown that the damping behavior is described much better than by other models with comparably few parameters. Moreover, it is the only one that works in a wide frequency range and can describe the dispersion of the resonance frequencies. The implementation of this damping description in a Boundary Element Code is an application of dynamics of 3D continua in the frequency domain.

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