Efficient Numerical Algorithm for the Solution of Nonlinear Two-Dimensional Volterra Integral Equation Arising from Torsion Problem

In this article, an effective method is given to solve nonlinear two-dimensional Volterra integral equations of the second kind, which is arising from torsion problem for a long bar that consists of the nonlinear viscoelastic material type with a fixed elliptical cross section. First, the existence of a unique solution of this problem is discussed, and then, we find the solution of a nonlinear two-dimensional Volterra integral equation (NT-DVIE) using block-by-block method (B-by-BM) and degenerate kernel method (DKM). Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the method.


Introduction
The equations of the torsion problem were derived in detail with analytical solutions, by Muskhelishvili [1], Frank and Mises [2], Nowinski [3], and Sneddon and Berry [4]. The problem can be formulated as a boundary value problem of the Laplace equation. In [5], boundary element method was developed for the nonuniform torsion of simply or multiply connected cylindrical bars of arbitrary cross section, where the bar is subjected to an arbitrary distributed twisting moment while its edges are restrained by the most general linear torsional. In [6], nonlinear inelastic uniform torsion of bars by BEM was studied. Sapountzakis and Tsipiras in [7] used the boundary element method solution to the nonlinear inelastic uniform torsion problem of composite bars. El-Kalla and AL-Bugami in [8] discussed the nonlinear Volterra-Fredholm integral equation and torsion problems. Sheshtawy and Ghaleb in [9], discussed approximate solution to the problem of torsion by a boundary integral method. Assari, in [10], discussed the numerical solution of T-DFIE of the second kind on nonrectangular domains. Fattahzadeh, in [11], solved two-dimensional linear and nonlinear Fredholm integral equations of the first kind based on Haar wavelet. Authors, in [12], solved twodimensional integral equation of the first kind by a multistep method. Alturk, in [13], solved two-dimensional Fredholm integral equations of the first kind using regularization-homotopy method. In this work, effective numerical methods are proposed to obtain the solution of nonlinear two-dimensional Volterra integral equations of the second kind and study the values of absolute errors.

Basic Formulas
While one end of the bar, of length b, is prevented from rotating, the other end is rotated about the z-axis. So that a section at distancezfrom the fixed end turns through angleθ, the variation of angleθwithz,z ∈ ½0, b, is taken as α is a twist angle. The displacement u θ of a particle in a tangential direction is given by where r is the radius of the particle. Then, we get From (1), (2), and (3), we obtained A comparison of (6) and (8) now gives The functions θ and ψ must satisfy the relations Stresses are derived from scalar ϕ in (6) in such a way that rectangular axes with any orientation may be used.
Let the origin of coordinate axes (n, s) be situated on the boundary of the section, direction n being normal to the boundary and directions being tangential to it. Local values of stresses are now given by The force p x acting on a vertical strip of width δx is given by It can be seen that the torque on the section is given by where T is the moment of torque.

Solution in the Form of NTVIE
The deviator strain of nonlinear elastic material is as follows: Using (5) in (14), we get Advances in Mathematical Physics The strain deviator tensor is defined as E ij = e ij -1 3 e kk δ ij , e kk = e xx + e yy + e zz , i, j = x, y, z, The second invariant of strain and stress tensor is as follows: Using (5) in (17), we get Using (17) again, we get Therefore, for the stress components, we find that σ xz , σ yz are the only nonvanishing components of stress; thus, we find Also, In addition, the principal cubic theory is given by where G is the shear modulus of the material and Jðt − τ, x − yÞ, kðt − τ, x − yÞ are the kernel functions. From (21) and (22), Using (15) and (19) in (22), (23), and (24), we obtain Also, Using (25) and (26) in (34), we have Let Then, equation (27) becomes Then, we get where a and b are the semimajor axis of the ellipse.

Advances in Mathematical Physics
By calculating∂ψ/∂x and ∂ψ/∂yfrom equation (31) and introducing the result in (29), we find Also, Then, we have Here, A 1 is the torsional rigidity and A 2 is the polar moment of inertia of the cross section of the bar.
If the bar is a linear and viscoelastic materialkðjt − τj, jx − yjÞ, then we get If the bar is a nonlinear and viscoelastic material, we get Jðjt − τj, jx − yjÞ = 0; then, The general form of formula (36) is where- , T, are given continuous functions and αðx, tÞ is an unknown function. λ 1 , λ 2 known constants, which have many physical meaning, may be complex. Jðjt − τj, jx − yjÞ and kðjt − τj, jx − yjÞ are continuous.

The Existence of a Unique Solution of T-DVIE
To discuss the existence and uniqueness solution of equation (39), we write it in the integral operator form where In addition, we assume the following conditions: (2) f ðx, tÞ, with its partial derivatives with respect to x and t, is continuous in L 2 ½0, b × C½0, T, and its norm is defined as (3) The known continuous functionγðy, τ, αðy, τÞÞ satisfies, for the constants A > A 1 , A > P, the following 4 Advances in Mathematical Physics conditions: (4) The unknown function αðx, tÞ satisfies the Lipchitz condition for the first argument of position and H€ o lder condition for the second argument of time, where (5) The kernels satisfies the Lipchitz condition with respect to position and H€ older condition with respect to time, where Proof. In the light of the two formulas (40) and (42) ☐ Using conditions (1) and (2), then applying Cauchy-Schwarz inequality, we get In view of condition (3-a), the above inequality takes the form Inequality (49) shows that the operator Q maps the ball Since ρ > 0 and G > 0, therefore we have σ < 1. Moreover, the inequality (49) involves the boundedness of the operator Q of equation (42), where In addition, the inequalities (49) and (51), define the boundedness of the operator Q. (1) and (3-b) are verified, and then, Q is a contraction operator in the space

Lemma 3. Assume that the conditions
Proof. For α 1 ðx, tÞ and α 2 ðx, tÞ in the space L 2 ½0, b × C½0, T and from equations (40) and (42), we find ☐ With the aid of conditions (1) and (3-b), the above inequality becomes Then, we get From inequality (54), we see that Q is continuous in the space L 2 ½0, b × C½0, T, and then, Q is a contraction operator under the condition σ < 1.

Solution of NT-DVIE
We consider the bar in the nonlinear case; then, the integral equation (39) with continuous kernel reduced to where γðy, τ, αðy, τÞÞand f ðx, tÞ ∈ L 2 ½0, b × C½0, T are given continuous functions. λ, which have many physical meaning, may be complex. The kernel kðjt − τj, jx − yjÞ is continuous.

The B-by-BM.
In this section, we use the B-by-BM for solving the NT-DVIE of the second kind. The interval ½0, b is divided into steps of width h, x j = j h, j = 0, 1, ⋯, n, and h = ðb − aÞ/n. The approximate solution of α 1 ðxÞ will be defined at mesh points x j and denoted by α ij , j = 0, 1, ⋯, n, such as α ij is an approximation to α i ðx j Þ.
To solve the NT-DVIE,              has been presented to solution the problem. These methods have proven to be effective in solving an equation NT-DVIE. Error analysis and some numerical examples are presented for different materials to illustrate the effectiveness and accuracy of the methods.
From the previous results in Tables 1 and 2 and Figures 1-12, we notice the following: (1) When the values of υ and λ are fixed in the linear and nonlinear case, then the error value increases with the time x, t = 0:2, 0.6, 1 (2) In the linear and nonlinear case, when the values of time are fixed, the error value increases with the increase of υ and λ (3) When the values of υ, λ, and time t are fixed, the error value decreases with N which is increasing, for the linear and nonlinear case and for each material (plutonium, steel, and copper) (4) As x is increasing and t is fixed, the errors are also increasing for the linear and nonlinear case and for each material (5) The approximate solutions calculated by B-by-BM and DKM are best methods for LT-DVIE and NT-DVIE (6) In general, the maximum value of the errors by B-by-BM and DKM in the linear case is less than the maximum value of the errors in the nonlinear case, for all materials, and the minimum value of the errors in the linear case is larger than the minimum value of the errors in the nonlinear case (7) The previous numerical experiments illustrate the accuracy of the proposed methods to solve the problem

Data Availability
All the data are available within the article and also as the references that were cited.

Conflicts of Interest
The author declares that there are no conflicts of interest.